Operadores de Calderón-Zygmund e o teorema T1

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❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙

❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊ ❚❊❈◆❖▲❖●■❆

P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆

❖♣❡r❛❞♦r❡s ❞❡ ❈❛❧❞❡ró♥✲❩②❣♠✉♥❞ ❡ ♦ ❚❡♦r❡♠❛ ❚✶

❘♦①❛♥❛ ❇❡❞♦②❛ Pr❛❞♦

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❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙

❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊ ❚❊❈◆❖▲❖●■❆

P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆

❖♣❡r❛❞♦r❡s ❞❡ ❈❛❧❞❡ró♥✲❩②❣♠✉♥❞ ❡ ♦ ❚❡♦r❡♠❛ ❚✶

❘♦①❛♥❛ ❇❡❞♦②❛ Pr❛❞♦

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦

PP●✲▼ ❞❛ ❯❋❙❈❛r ❝♦♠♦ ♣❛rt❡

❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦

❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛✲

t❡♠át✐❝❛✳

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ▲✉ís ❆♥tô♥✐♦ ❈❛r✈❛❧❤♦ ❞♦s ❙❛♥t♦s

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Ficha catalográfica elaborada pelo DePT da Biblioteca Comunitária da UFSCar

P896oc

Prado, Roxana Bedoya.

Operadores de Calderón-Zygmund e o teorema T1 / Roxana Bedoya Prado. -- São Carlos : UFSCar, 2009. 136 f.

Dissertação (Mestrado) -- Universidade Federal de São Carlos, 2009.

1. Análise harmônica. 2. Integrais Singulares. 3. BMO. 4. Fourier, Análise de. I. Título.

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Banca Examinadora:

DM

-

UFSCar

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❆❣r❛❞❡❝✐♠❡♥t♦s

❆ ❉❡✉s ♣❡❧❛ ❝♦♥st❛♥t❡ ❝♦♠♣❛♥❤✐❛ ❡ ♣♦r ❡st❛r s❡♠♣r❡ ✐❧✉♠✐♥❛♥❞♦ ♠❡✉ ❝❛♠✐♥❤♦✳

❆♦s ♠❡✉s ♣❛✐s✱ ♣♦r t♦❞♦ ❛♠♦r✱ ❝❛r✐♥❤♦✱ ❝♦♠♣r❡♥sã♦✱ ❛♣♦✐♦ ❡ ✐♥❝❡♥t✐✈♦ ❡♠ ♠❡✉s ❡st✉❞✐♦s✳ ❆♦ ♣r♦❢❡ss♦r ▲✉ís ❆♥tô♥✐♦ ❈❛r✈❛❧❤♦ ❞♦s ❙❛♥t♦s✱ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ♣❡❧❛ ♦r✐❡♥t❛çã♦ ❡ ❞❡❞✐✲ ❝❛çã♦ ❛ ❡st❡ tr❛❜❛❧❤♦✱ ♣❡❧❛ ❝♦♥✜❛♥ç❛✱ ❛♠✐③❛❞❡✱ ❜♦♠ ❤✉♠♦r ❡ ♣❡❧❛ ♣r❡♦❝✉♣❛çã♦ q✉❡ s❡♠♣r❡ ❞❡♠♦♥str♦✉ ♣♦r ♠✐♠✳

❆♦s ♣r♦❢❡s♦r❡s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋❙❈❛r ♣❡❧♦s ❡♥s✐♥❛♠✐❡♥t♦s ♠❛✲ t❡♠át✐❝♦s ❡ ♣❡❧❛ ❛♠✐③❛❞❡✳

❆♦s ❛♠✐❣♦s ❞♦ ❉▼ ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ✐♥❝❡♥t✐✈♦✳ ❊ á ❈❆P❊❙ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳

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❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦✱ ❛♣r❡s❡♥t❛♠♦s ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ❡ s✉✜❝✐❡♥t❡s ♣❛r❛ q✉❡ ✉♠ ♦♣❡r❛❞♦r ❞❡ t✐♣♦ ❈❛❧❞❡ró♥✲❩②❣♠✉♥❞ ♣♦ss❛ s❡r ❡st❡♥❞✐❞♦ ❛ ✉♠ ♦♣❡r❛❞♦r ❧✐♠✐t❛❞♦ ❡♠ L2_{✳ ❯s❛♥❞♦}

r❡s✉❧t❛❞♦s ❞❡ ✐♥t❡r♣♦❧❛çã♦ ❡ ❝♦♥❞✐çõ❡s ❞❡ ❝❛♥❝❡❧❛♠❡♥t♦s s♦❜r❡ ♦ ♥ú❝❧❡♦ ♦❜t❡r ❛ ❧✐♠✐t❛çã♦ ❡♠ Lp_{✱ ♣❛r❛ t♦❞♦} ₁_{< p <}_∞_✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❆♥á❧✐s❡ ❍❛r♠ó♥✐❝❛✱ ■♥t❡❣r❛✐s ❙✐♥❣✉❧❛r❡s✱ ❖♣❡r❛❞♦r❡s ❞❡ ❈❛❧❞❡ró♥✲❩②❣♠✉♥❞✱ ❚❡♦r❡♠❛ ❚✶✳

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❆❜str❛❝t

■♥ t❤✐s ✇♦r❦✱ ✇❡ ♣r❡s❡♥t ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s ❢♦r t❤❛t ❛♥ ♦♣❡r❛t♦r ♦❢ ❈❛❧❞❡ró♥✲ ❩②❣♠✉♥❞ t②♣❡ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ❛ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦r ♦♥ L2_{✳ ❇② ✉s✐♥❣ r❡s✉❧ts ♦❢ ✐♥t❡r✲}

♣♦❧❛t✐♦♥ ❛♥❞ ❝❛♥❝❡❧❧❛t✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ ❦❡r♥❡❧ t♦ ♦❜t❛✐♥ t❤❡ ❜♦✉♥❞❡♥❡ss ✐♥ Lp_{✱ ❢♦r ❛❧❧}

1< p <_∞✳

❑❡②✇♦r❞s✿ ❍❛r♠♦♥✐❝ ❆♥❛❧②s✐s✱ ❙✐♥❣✉❧❛r ■♥t❡❣r❛❧s✱ ❈❛❧❞❡ró♥✲❩②❣♠✉♥❞ ❖♣❡r❛t♦rs✱ t❤❡ ❚✶ t❤❡♦r❡♠✳

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❙✉♠ár✐♦

✶ Pr❡❧✐♠✐♥❛r❡s ✶✷

✶✳✶ ❚❡♦r✐❛ ❞❛s ❉✐str✐❜✉✐çõ❡s ❡ ❆♥á❧✐s❡ ❞❡ ❋♦✉r✐❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✶✳✶✳✶ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✶✳✶✳✷ ❆ ❈❧❛ss❡ ❞❛s ❉✐str✐❜✉✐çõ❡s ❚❡♠♣❡r❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✶✳✷ ❋✉♥çã♦ ▼❛①✐♠❛❧ ❞❡ ❍❛r❞②✲▲✐tt❧❡✇♦♦❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✶✳✸ ❚❡♦r❡♠❛ ❞❡ ■♥t❡r♣♦❧❛çã♦ ❞❡ ▼❛r❝✐♥❦✐❡✇✐❝③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✶✳✹ ❚❡♦r❡♠❛ ❞❡ ❉✐❢❡r❡♥❝✐❛çã♦ ❞❡ ❘✐❡♠❛♥♥✲▲❡❜❡s❣✉❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

✶✳✺ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❈❛❧❞❡ró♥✲❩②❣♠✉♥❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽

✶✳✻ ❆ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❍✐❧❜❡rt✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✷ ■♥t❡❣r❛✐s ❙✐♥❣✉❧❛r❡s ✹✸

✷✳✶ ❉✐str✐❜✉✐çõ❡s ❍♦♠♦❣ê♥❡❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✷✳✷ ❖ ▼ét♦❞♦ ❞❛s ❘♦t❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

✷✳✷✳✶ ❖♣❡r❛❞♦r❡s ❘❡♣r❡s❡♥t❛❞♦s ♣♦r ◆ú❝❧❡♦ ❮♠♣❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

✷✳✷✳✷ ❚r❛♥s❢♦r♠❛❞❛s ❞❡ ❘✐❡s③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

✷✳✷✳✸ ❖♣❡r❛❞♦r❡s ❘❡♣r❡s❡♥t❛❞♦s ♣♦r ◆ú❝❧❡♦ P❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵

✷✳✸ ■♥t❡❣r❛✐s ❙✐♥❣✉❧❛r❡s ❞♦ ❚✐♣♦ ❈♦♥✈♦❧✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻

✷✳✹ ■♥t❡❣r❛✐s ❚r✉♥❝❛❞❛s ❡ ♦ ❱❛❧♦r Pr✐♥❝✐♣❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸

✸ ❖♣❡r❛❞♦r❡s ❞❡ ❈❛❧❞❡ró♥✲❩②❣♠✉♥❞ ✽✶

✸✳✶ ■♥t❡❣r❛✐s ❙✐♥❣✉❧❛r❡s ❞❡ ❈❛❧❞❡ró♥✲❩②❣♠✉♥❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶

✹ ❖ ❊s♣❛ç♦ ❇▼❖ ✾✽

✹✳✶ ❖ ❊s♣❛ç♦ ❆tô♠✐❝♦ H1 _{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✽}

✹✳✷ ❖ ❊s♣❛ç♦ BM O✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✶

✹✳✷✳✶ Pr♦♣r✐❡❞❛❞❡s ❞❛ ❋✉♥çã♦ ❙❤❛r♣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✻

✹✳✷✳✷ ❆ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❏♦❤♥✲◆✐r❡♥❜❡r❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✶

✺ ❖ ❚❡♦r❡♠❛ ❚✶ ✶✶✻

✺✳✶ ❊①♣♦s✐çã♦ ❡ ❆♣❧✐❝❛çõ❡s ❞♦ ❚❡♦r❡♠❛ ❚✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✾

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❚á❜✉❛ ❞❡ ◆♦t❛çã♦

C∞

c ✿ ❢✉♥çõ❡s ✐♥✜♥✐t❛♠❡♥t❡ ❞✐❢❡r❡♥❝✐á✈❡✐s ❝♦♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦

S ✿ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ❙❝❤✇❛rt③

S′ _{✿ ❡s♣❛ç♦ ❞❛s ❞✐str✐❜✉✐çõ❡s t❡♠♣❡r❛❞❛s}

F,_b ✿ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r

F−1 _{✿ ✐♥✈❡rs❛ ❞❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r}

f ∗g ✿ ❝♦♥✈♦❧✉çã♦ ❞❛s ❢✉♥çõ❡s f ❡ g | | ✿ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡

s✉♣♣(f) ✿ s✉♣♦rt❡ ❞❡ f

M f ✿ ❢✉♥çã♦ ♠❛①✐♠❛❧ ❞❡ ❍❛r❞② ▲✐tt❧❡✇♦♦❞ Pt(x) ✿ ♥ú❝❧❡♦ ❞❡ P♦✐ss♦♥

Qt(x) ✿ ❝♦♥❥✉❣❛❞♦ ❤❛r♠ô♥✐❝♦ ❞❡ Pt(x)

✈✳♣✳ K ✿ ✈❛❧♦r ♣r✐♥❝✐♣❛❧ ❞❛ ❞✐str✐❜✉✐çã♦ t❡♠♣❡r❛❞❛ K Hf ✿ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❍✐❧❜❡rt

Rjf ✿ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❘✐❡s③

h , i ✿ ❡♠♣❛r❡❧❤❛♠❡♥t♦ ❡♥tr❡ ❞✐str✐❜✉✐çõ❡s t❡♠♣❡r❛❞❛s ❡ ❢✉♥çõ❡s ❞♦ ❡s♣❛❝♦ ❞❡ ❙❝❤✇❛rt③ T∗ _{✿ ♦♣❡r❛❞♦r ❛❞❥✉♥t♦}

H1

at ✿ ❡s♣❛ç♦ ❛tô♠✐❝♦ H1

H1 _{✿ ❡s♣❛ç♦ ❞❡ ❍❛r❞②}

BM O ✿ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ❞❡ ♦s❝✐❧❛çã♦ ♠é❞✐❛ ❧✐♠✐t❛❞❛

(11)

■♥tr♦❞✉çã♦

❆s ✐♥t❡❣r❛✐s s✐♥❣✉❧❛r❡s sã♦ ❝❡♥tr❛✐s ♣❛r❛ ❛ ❛♥á❧✐s❡ ❤❛r♠ô♥✐❝❛ ❛❜str❛❝t❛✱ ❡ sã♦ ✐♥t✐♠❛♠❡♥t❡ ❝♦♥❡❝t❛❞❛s ❝♦♠ ♦ ❡st✉❞♦ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s✳ ❆ ✐♥t❡❣r❛❧ s✐♥❣✉❧❛r é ✉♠ ♦♣❡r❛❞♦r ✐♥t❡❣r❛❧

T f(x) =

Z

K(x, y)f(y)dy ✭✵✳✵✳✶✮ ❝✉❥❛ ❢✉♥çã♦ ♥ú❝❧❡♦ K : Rn _×_Rn _−→ _C _{é s✐♥❣✉❧❛r ❛♦ ❧♦♥❣♦ ❞❛ ❞✐❛❣♦♥❛❧} _x ₌ _y_{✳ ❈♦♠♦}

t❛✐s ✐♥t❡❣r❛✐s ♥ã♦ ♣♦❞❡♠ ❡♠ ❣❡r❛❧ s❡r ❛❜s♦❧✉t❛♠❡♥t❡ ✐♥t❡❣rá✈❡✐s✱ ✉♠❛ ❞❡✜♥✐çã♦ r✐❣♦r♦s❛ ❞❡✈❡ ❞❡✜♥í✲❧❛s ❝♦♠♦ ♦ ❧✐♠✐t❡ ❞❛ ✐♥t❡❣r❛❧ s♦❜r❡ |x₋y_| > ǫ q✉❛♥❞♦ ǫ _→ 0✳ ❯s✉❛❧♠❡♥t❡

s✉♣♦s✐çõ❡s ❛❞✐❝✐♦♥❛✐s sã♦ r❡q✉❡r✐❞❛s ♣❛r❛ ♦❜t❡r r❡s✉❧t❛❞♦s t❛❧ ❝♦♠♦ s✉❛ ❧✐♠✐t❛çã♦ ❡♠Lp₍_Rn₎_✱

1 < p < _∞✳ ❉✐③❡♠♦s q✉❡ ♦ ♥ú❝❧❡♦ K : Rn_×_Rn_\_∆ _−→ _C _{é ✉♠ ♥ú❝❧❡♦ ♣❛❞rã♦ s❡ ❡①✐st❡}

δ >0 t❛❧q✉❡

|K(x, y)_{| ≤} C

|x₋y_|n,

|K(x, y)₋K(x, z)_{| ≤}C |y−z|

δ

|x₋y_|n+δ s❡ |x−y|>2|y−z|,

|K(x, y)₋K(w, y)_{| ≤}C |x−w|

δ

|x₋y_|n+δ s❡ |x−y|>2|x−w|.

❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡st❡ tr❛❜❛❧❤♦ é ❞❡t❡r♠✐♥❛r ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ❡ s✉✜❝✐❡♥t❡s ♣❛r❛ q✉❡ ✉♠ ♦♣❡r❛❞♦r✱ ❝✉❥♦ ♥ú❝❧❡♦ é ♣❛❞rã♦✱ s❡❥❛ ❡st❡♥❞✐❞♦ ❛ ✉♠ ♦♣❡r❛❞♦r ❧✐♠✐t❛❞♦ ❡♠ L2_✳

❖✉ s❡❥❛✱ ❞❡t❡r♠✐♥❛r ❛s ❝♦♥❞✐çõ❡s ❛❞✐❝✐♦♥❛✐s s♦❜r❡ T ♣❛r❛ ❣❛r❛♥t✐r q✉❡ ❡st❡ s❡ ❡①t❡♥❞❛ ❛ ✉♠ ♦♣❡r❛❞♦r ❧✐♠✐t❛❞♦ ❞❡ L2 _{❡♠ s✐ ♠❡s♠♦❀ ❛❧é♠ ❞✐ss♦✱ s✉♣♦♥❞♦ q✉❡} _K _{é ✉♠ ♥ú❝❧❡♦}

♣❛❞rã♦✱ ❞❡t❡r♠✐♥❛r ❛s ❝♦♥❞✐çõ❡s ❛❞✐❝✐♦♥❛✐s s♦❜r❡ K t❛❧ q✉❡ ❡①✐st❛ ✉♠ ♦♣❡r❛❞♦r ❧✐♠✐t❛❞♦ T :L2₍_Rn₎_→_L2₍_Rn₎ _{t❡♥❞♦ ♥ú❝❧❡♦}_K _{♥♦ s❡♥t✐❞♦ ❞❡ ✭}_{✵✳✵✳✶}_✮✳

❖ ♦♣❡r❛❞♦r ✐♥t❡❣r❛❧ s✐♥❣✉❧❛r ♠❛✐s ❝♦♥❤❡❝✐❞♦ é ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❍✐❧❜❡rtH✳ ❊st❡ é ❞❛❞♦ ♣♦r ❝♦♥✈♦❧✉çã♦ ❝♦♥tr❛ ♦ ♥ú❝❧❡♦ K(x) = 1/(πx) ♣❛r❛ x∈R✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱

Hf(x) = 1

π limǫ→0 Z

|x−y|>ǫ

1

x₋yf(y)dy.

P❛r❛ ♦♣❡r❛❞♦r❡s ♠❛✐♦r❡s q✉❡ ✶✱ ♦s ♦♣❡r❛❞♦r❡s ❛♥á❧♦❣♦s ❛ ❡st❡ sã♦ ❛s tr❛♥s❢♦r♠❛❞❛s ❞❡ ❘✐❡s③✱ ♦ q✉❛❧ s✉❜st✐t✉✐♠♦s K(x) = 1/x ❝♦♠ Ki(x) = _|_x_|xni+1✱ ❛ss✐♠

Rjf(x) = cn✈✳♣✳

Z

Rn yj

|y_|n+1f(x−y)dy, 1≤j ≤n,

(12)

s❡♥❞♦ xi é ❛ ✐✲és✐♠❛ ❝♦♠♣♦♥❡♥t❡ ❞❡ x∈Rn✳

◆♦ ❝❛♣ít✉❧♦ ✶ sã♦ ❢❡✐t❛s ❛❧❣✉♠❛s ♣r❡❧✐♠✐♥❛r❡s ♥❡❝❡ssár✐❛s ❛♦ ❜♦♠ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ❡ sã♦ ❞❡♠♦♥str❛❞♦s ♦ t❡♦r❡♠❛ ❞❡ ✐♥t❡r♣♦❧❛çã♦ ❞❡ ▼❛r❝✐♥❦✐❡✇✐❝③ ❡ ❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ❈❛❧❞❡ró♥✲❩②❣♠✉♥❞✱ ♦s q✉❛✐s sã♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♠♦str❛r q✉❡ ✉♠❛ ✐♥t❡❣r❛❧ s✐♥❣✉❧❛r ❝♦♠ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s é ❞❡ t✐♣♦ ❢r❛❝❛(1,1)❡ ❧✐♠✐t❛❞❛ ❡♠Lp_{✳ ◆❡st❡ ❝❛♣ít✉❧♦ s❡ ♠♦str❛}

q✉❡ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❍✐❧❜❡rt é ❞❡ t✐♣♦ ❢r❛❝❛(1,1)❡ ❧✐♠✐t❛❞❛ ❡♠ Lp_✳

◆♦ ❝❛♣ít✉❧♦ ✷ ❡st✉❞❛♠♦s ❛s ✐♥t❡❣r❛✐s s✐♥❣✉❧❛r❡s ❞❛ ❢♦r♠❛ T f(x) = lim

ǫ→0 Z

|y|>ǫ

Ω(y′₎

|y_|n f(x−y)dy ✭✵✳✵✳✷✮

❝♦♠ Ω _∈ L1₍_Sn−1₎_✱ R

Sn−1Ω(u)dσ(u) = 0 ❡ y′ = y/|y|✳ P❡❧♦ ♠ét♦❞♦ ❞❛s r♦t❛çõ❡s ❡ ❞❛s

❚r❛♥s❢♦r♠❛❞❛s ❞❡ ❘✐❡s③ ♠♦str❛♠♦s q✉❡ s❡ Ω é ✉♠❛ ❢✉♥çã♦ ❡♠ Sn−1 _{❝♦♠ ♠é❞✐❛ ③❡r♦ t❛❧}

q✉❡ s✉❛ ♣❛rt❡ í♠♣❛r ❡stá ❡♠ L1₍_Sn−1₎ _{❡ s✉❛ ♣❛rt❡ ♣❛r ❡stá ❡♠} _Lq₍_Sn−1₎_{♣❛r❛ ❛❧❣✉♠} _{q >}₁_✱

❡♥tã♦ ❛ ✐♥t❡❣r❛❧ s✐♥❣✉❧❛r T ❡♠ ✭✵✳✵✳✷✮ é ❧✐♠✐t❛❞♦ ❡♠ Lp₍_Rn₎_✱ ₁_{< p <}_∞_✳

❆❧é♠ ❞✐ss♦✱ s❡ ❛❞✐❝✐♦♥❛♠♦s ❝♦♥❞✐çõ❡s s♦❜r❡ ♦ ♥ú❝❧❡♦K _∈L1

loc(Rn\{0})❝♦♠♦ ❛ ❝♦♥❞✐çã♦

❞♦ t❛♠❛♥❤♦

sup

R>0 Z

R<|x|<2R|

K(x)_|dx_≤C,

❛ ❝♦♥❞✐çã♦ ❞❡ s✉❛✈✐❞❛❞❡

sup

y6=0 Z

|x|>2|y||

K(x₋y)₋K(x)_|dx_≤C

❡ ❛ ❝♦♥❞✐çã♦ ❞❡ ❝❛♥❝❡❧❛çã♦

Z

R1<|x|<R2

K(x)dx= 0, ♣❛r❛ t♦❞♦ R1, R2 >0.

❊♥tã♦T = lim

ǫ→0 Z

|y|>ǫ

K(y)f(x−y)dyé ❧✐♠✐t❛❞♦ ❡♠Lp(Rn₎_{❡ s❛t✐s❢❛③ ❛ ❡st✐♠❛t✐✈❛ t✐♣♦ ❢r❛❝❛}

(1,1)✳ ❖❜s❡r✈❡ q✉❡ ❡st❛s ❝♦♥❞✐çõ❡s sã♦ s❛t✐s❢❡✐t❛s ♣❡❧❛s tr❛♥s❢♦r♠❛❞❛s ❞❡ ❍✐❧❜❡rt ❡ ❞❡ ❘✐❡s③✱

❛ss✐♠ ❡st❡ r❡s✉❧t❛❞♦ é ✉♠❛ ❡①t❡♥sã♦ ❞❡ ❛q✉❡❧❡s r❡s✉❧t❛❞♦s✳

❖s ♦♣❡r❛❞♦r❡s ❡st✉❞❛❞♦s ♥♦s ❝❛♣✐t✉❧♦s ✶ ❡ ✷ ♣♦❞❡♠ s❡r ❡s❝r✐t♦s ❝♦♠♦ ❝♦♥✈♦❧✉çõ❡s ❝♦♠ ❞✐str✐❜✉✐çõ❡s t❡♠♣❡r❛❞❛s✳ ❉✐③✲s❡ q✉❡T é ✉♠ ♦♣❡r❛❞♦r ❞❡ ❈❛❧❞❡ró♥✲❩②❣♠✉♥❞ s❡T é ❧✐♠✐t❛❞♦ ❡♠ L2 _{❡ ❡①✐st❡ ✉♠ ♥ú❝❧❡♦ ♣❛❞rã♦} _K _{t❛❧ q✉❡ ♣❛r❛} _f _∈_L2 _{❝♦♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦✱}

T f(x) =

Z

Rn

K(x, y)f(y)dy, x /_∈s✉♣♣(f).

◆♦ ❝❛♣✐t✉❧♦ ✸ ❡st✉❞❛♠♦s ✐♥t❡❣r❛✐s s✐♥❣✉❧❛r❡s ❞♦ t✐♣♦ ♥ã♦ ❝♦♥✈♦❧✉çã♦✱ ♠♦str❛♠♦s q✉❡ ✉♠ ♦♣❡r❛❞♦r ❞❡ ❈❛❧❞❡ró♥✲❩②❣♠✉♥❞ é ❧✐♠✐t❛❞♦ ❡♠ Lp_✱ ₁ _{< p <}_∞_{✱ ❡ é ❢r❛❝❛} ₍₁_,₁₎_{✳ ❆❧é♠ ❞✐ss♦}

❞❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ♦s q✉❛✐s tê♠ ♥ú❝❧❡♦ ♣❛❞rã♦ ❝♦♠♦✿ ❛ ✐♥t❡❣r❛❧ ❞❡ ❈❛✉❝❤② ❛♦ ❧♦♥❣♦ ❞❡ ✉♠❛ ❝✉r✈❛ ▲✐♣s❝❤✐t③ ❡ ❈♦♠✉t❛❞♦r❡s ❞❡ ❈❛❧❞❡ró♥✳

◆♦ ❝❛♣ít✉❧♦ ✹ ❡st✉❞❛♠♦s ♦ ❡s♣❛ç♦ ❇▼❖(Rn₎ _{❡ sã♦ ❞✐s❝✉t✐❞♦s ❛❧❣✉♥s ❞❡ s❡✉s ♣r✐♥❝✐♣❛✐s}

r❡s✉❧t❛❞♦s ❡ ❡①❛♠✐♥❛♠♦s ❛ r❛③ã♦ ❞❡ ❝r❡❝✐♠❡♥t♦ ❞❡ ❢✉♥çõ❡s ❡♠ BM O ❝♦♠ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❏♦❤♥✲◆✐r❡♥❜❡r❣✳

(13)

❆ ❢✉♥çã♦ φ _∈ C∞

c (Rn) é ❞✐t❛ ❢✉♥çã♦ t❡st❡ ♥♦r♠❛❧✐③❛❞❛ s❡ s✉♣♣(φ) ⊆ B(0,1) ❡ ❡①✐st❡

N >0t❛❧ q✉❡ _k∂α_φ_k

L∞ ≤1✱ ♣❛r❛ t♦❞♦ |α| ≤N✳

❯♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r T é ❞✐t♦ r❡str✐t❛♠❡♥t❡ ❧✐♠✐t❛❞♦ s❡ ♣❛r❛ t♦❞❛φ ❢✉♥çã♦ t❡st❡ ♥♦r♠❛✲ ❧✐③❛❞❛✱ T(φx0,R₎∈_L2₍Rn₎❡ ❡①✐st❡ _{A >}₀ t❛❧ q✉❡ k_T₍_φx0,R₎k

L2 ≤A Rn/2 ♣❛r❛ t♦❞♦x₀ ∈Rn

❡ R >0✱ ❝♦♠A ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ x0, R, φ✳

❚❡r♠✐♥❛♠♦s ❡♥tã♦ ♥♦ss♦ tr❛❜❛❧❤♦ ❝♦♠ ♦ ❝❛♣ít✉❧♦ ✺✱ ♥♦ q✉❛❧ ❡st✉❞❛♠♦s ♥♦ss♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦✱ ♦ ❚❡♦r❡♠❛ T1✱ ♦ q✉❛❧ ♥♦s ❞✐③ q✉❡ ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❝♦♥tí♥✉♦ T : _{S → S}′

❛ss♦❝✐❛❞♦ ❝♦♠ ✉♠ ♥ú❝❧❡♦ ♣❛❞rã♦ K✱ s❡ ❡st❡♥❞❡ ❛ ✉♠ ♦♣❡r❛❞♦r ❧✐♠✐t❛❞♦ ❡♠ L2₍_Rn₎ _{s❡✱ ❡}

s♦♠❡♥t❡ s❡✱T ❡T∗ _{sã♦ r❡str✐t❛♠❡♥t❡ ❧✐♠✐t❛❞♦s✳ ❊ ❞❛♠♦s ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞❡st❡ t❡♦r❡♠❛✳}

(14)

❈❛♣ít✉❧♦ ✶

Pr❡❧✐♠✐♥❛r❡s

✶✳✶ ❚❡♦r✐❛ ❞❛s ❉✐str✐❜✉✐çõ❡s ❡ ❆♥á❧✐s❡ ❞❡ ❋♦✉r✐❡r

❊st❛ s❡çã♦ ❝♦♥té♠ r❡s✉❧t❛❞♦s ❞❡ t❡♦r✐❛ ❞❛s ❉✐str✐❜✉✐çõ❡s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s ♥♦s ❝❛♣ít✉❧♦s s✉❜s❡qü❡♥t❡s✳ ❆ ❜✐❜❧✐♦❣r❛✜❛ ❜ás✐❝❛ ✉t✐❧✐③❛❞❛ ♥❡st❛ s❡çã♦ é ❬❙❲❪✳

❈♦♠❡ç❛♠♦s ❝♦♥s✐❞❡r❛♥❞♦ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ♥♦s ❡s♣❛ç♦s L1₍_Rn₎ _❡ _L2₍_Rn₎_{✳ ❆ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❡ ❛s♣❡❝t♦ ❢♦r♠❛❧ é ♠❛✐s ❢❛❝✐❧♠❡♥t❡ ❞❡s❝r✐t❛}

♥♦ ❝♦♥t❡①t♦ ❞❡ ❞✐str✐❜✉✐çõ❡s❀ ♣♦rt❛♥t♦ ❡①t❡♥❞❡♠♦s s✉❛ ❞❡✜♥✐çã♦ ❛♦ ❡s♣❛ç♦ ❞❡ ❞✐str✐❜✉✐çõ❡s t❡♠♣❡r❛❞❛s✳

✶✳✶✳✶ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r

❆ ♦♣❡r❛çã♦ ❝♦♥✈♦❧✉çã♦ é ❞❡✜♥✐❞❛ ❛ss✐♠✿ ❙❡ f, g ∈ L1₍_Rn₎_{✱ s✉❛ ❝♦♥✈♦❧✉çã♦} _h ₌ _f _∗_g _{é ❛}

❢✉♥çã♦ ❝✉❥♦ ✈❛❧♦r ❡♠x_∈Rn _é

h(x) =

Z

Rn

f(x₋y)g(y)dy=

Z

Rn

f(y)g(x₋y)dy.

❉❛❞❛ ✉♠❛ ❢✉♥çã♦ f _∈L1₍_Rn₎_{✱ ❞❡✜♥❡ s✉❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ♣♦r}

ˆ

f(ξ) =

Z

Rn

f(x)e−2πix·ξdx,

♦♥❞❡ x·ξ

❖ s❡❣✉✐♥t❡ t❡♦r❡♠❛ é ✉♠❛ ❧✐st❛ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❞❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r✳ ❚❡♦r❡♠❛ ✶✳✶✳✶✳ ❛✮ kfˆ_kL∞₍Rn₎ ≤ kfk_L1 ❡ f é ❝♦♥tí♥✉❛✳

❜✮ lim_|ξ|→∞fˆ(ξ) = 0 ✭❘✐❡♠❛♥♥✲▲❡❜❡s❣✉❡✮✳

❝✮ (f_∗g)b_{= ˆ}_f_g✳_ˆ

(15)

❞✮ (τhf)∧(ξ) = e−2πih·ξfˆ(ξ)✱ ♦♥❞❡τhf(x) =f(x−h)❀ (e2πih·xf(x))∧(ξ) = ˆf(ξ−h)✳

❡✮ ❙❡ g(x) =λ−n_f₍_λ−1_x₎_{✱ ❡♥tã♦} _g_ˆ₍_ξ_{) = ˆ}_f₍_λξ₎_✳

❢✮ ❙❡ xjf(x)∈L1(Rn)✳ ❊♥tã♦ fˆé ❞✐❢❡r❡♥❝✐á✈❡❧ ❝♦♠ r❡s♣❡✐t♦ ❛ xj ❡

∂fˆ ∂ξj

(ξ) = (₋2πixjf)b(ξ).

❣✮ ❙❡ ∂f ∂xj ∈L

1₍_Rn₎ _❡♥tã♦

∂f ∂xj

b

(ξ) = 2πiξjfˆ(ξ).

❚❡♦r❡♠❛ ✶✳✶✳✷✳ ❙❡ f ∈L1₍_Rn₎_∩_L2₍_Rn₎ _❡♥tã♦ _k_fˆ_k

L2₍_Rn₎=kfk_L2₍_Rn₎✳

❉❡♠♦♥str❛çã♦✿ ❬❙❲❪

❚❡♦r❡♠❛ ✶✳✶✳✸✳ ❆ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r é ✉♠ ♦♣❡r❛❞♦r ✉♥✐tár✐♦ ❡♠ L2₍_Rn₎_✳

❉❡♠♦♥str❛çã♦✿ ❬❙❲❪

❚❡♦r❡♠❛ ✶✳✶✳✹✳ ❆ ✐♥✈❡rs❛ F−1 _{❞❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ♣♦❞❡ s❡r ♦❜t✐❞❛ ❞❡✜♥✐♥❞♦✲s❡}

(_F−1g)(x) = (_Fg)(₋x)

♣❛r❛ t♦❞❛ g ∈L2₍_Rn₎_✳

❉❡♠♦♥str❛çã♦✿ ❬❙❲❪

❚❡♦r❡♠❛ ✶✳✶✳✺✳ ❙❡ f _∈L1₍_Rn₎ _❡ _g _∈_Lp₍_Rn₎_✱ ₁_≤_p_≤₂_{✱ ❡♥tã♦} _h₌_f_∗_g _∈_Lp₍_Rn₎ _❡

(Fh)(x) = (Ff)(x)(Fg)(x)

♣❛r❛ q✉❛s❡ t♦❞♦ x∈Rn_✳

❉❡♠♦♥str❛çã♦✿ ❬❙❲❪

(16)

✶✳✶✳✷ ❆ ❈❧❛ss❡ ❞❛s ❉✐str✐❜✉✐çõ❡s ❚❡♠♣❡r❛❞❛s

❆ ✐❞é✐❛ ❜ás✐❝❛ q✉❡ ♠♦t✐✈❛ ❛ t❡♦r✐❛ ❞❛s ❞✐str✐❜✉✐çõ❡s ❡stá ❡♠ ❝♦♥s✐❞❡rá✲❧❛s ❝♦♠♦ ❢✉♥❝✐♦♥❛✐s ❧✐♥❡❛r❡s ❞❡✜♥✐❞♦s s♦❜r❡ ❛❧❣✉♠ ❡s♣❛ç♦ ❞❡ ❢✉♥çõ❡s r❡❣✉❧❛r❡s✱ ♦ ❝❤❛♠❛❞♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s t❡st❡s✳ ❖ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s t❡st❡ t❡♠ ❝♦♠♦ ♣r❡ss✉♣♦st♦ s❡r ❜❡♠ ❝♦♠♣♦rt❛❞♦ ❝♦♠ r❡✲ s♣❡✐t♦ ❛s ♦♣❡r❛çõ❡s✿ ❞✐❢❡r❡♥❝✐❛çã♦✱ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❋♦✉r✐❡r✱ ❝♦♥✈♦❧✉çã♦✱ tr❛♥s❧❛çã♦✱ ❡t❝✳ ❆ ❡s❝♦❧❤❛ ❞❛ t♦♣♦❧♦❣✐❛ ❞♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s t❡st❡ ♣❡r♠✐t❡ ❛ ❡①t❡♥sã♦ ♥❛t✉r❛❧ ❞❡st❛s ♦♣❡✲ r❛çõ❡s ❛♦ ❡s♣❛ç♦ ❞❛s ❞✐str✐❜✉✐çõ❡s✳ ❚❛✐s ♦♣❡r❛çõ❡s ♣❡r♠✐t❡♠ ♦ ❡st✉❞♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s ❞✐str✐❜✉✐çõ❡s✳

❉❡✜♥✐çã♦ ✶✳✶✳✶✳ ❉❡♥♦t❛♠♦s ❝♦♠ S(Rn₎ _{♦ s✉❜❡s♣❛ç♦ ❞❡} _C∞₍_Rn₎ _{❞❛s ❢✉♥çõ❡s} _φ _{t❛✐s q✉❡}

sup

x∈Rn

xαDβφ(x)<∞ ✭✶✳✶✳✶✮

♣❛r❛ t♦❞♦ α, β ∈ Nn_{✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ s❡q✉ê♥❝✐❛} _φ

j ∈ S(Rn)✱ j = 1,2, . . . ❝♦♥✈❡r❣❡ ♣❛r❛

③❡r♦ ❡♠ S(Rn₎_{✱ s❡ ♣❛r❛ t♦❞♦} _{α, β} _∈_Nn

sup

x∈Rn

xα_Dβ_φ j(x)

_→0, q✉❛♥❞♦, j _{→ ∞}.

❉❡✜♥✐çã♦ ✶✳✶✳✷✳ ❯♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❡ ❝♦♥tí♥✉♦ ❡♠ S(Rn₎ _{é ❞✐t♦ ✉♠❛ ❞✐str✐❜✉✐çã♦ t❡♠✲}

♣❡r❛❞❛✳ ❖ ❡s♣❛ç♦ ❞❛s ❞✐str✐❜✉✐çõ❡s t❡♠♣❡r❛❞❛s s❡ ❞❡♥♦t❛ ❝♦♠ S′_✳

❚❡♦r❡♠❛ ✶✳✶✳✻✳ ❙❡ φ _{∈ S} ❡♥tã♦ _Fφ_{∈ S}✳

❉❡♠♦♥str❛çã♦✿ ❬❙❲❪

❚❡♦r❡♠❛ ✶✳✶✳✼✳ ❙❡ φ, ψ∈ S ❡♥tã♦ φ∗ψ ∈ S✳

❉❡♠♦♥str❛çã♦✿ ❬❙❲❪

❆❧❣✉♠❛s ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ S ❡ ❞❡ s✉❛ t♦♣♦❧♦❣✐❛ ❡stã♦ ❡①♣r❡ss❛s ♥♦ s❡❣✉✐♥t❡ ❚❡♦r❡♠❛✳ ❚❡♦r❡♠❛ ✶✳✶✳✽✳ ❱❛❧❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣♦s✐çõ❡s

❛✮ ❆ ❛♣❧✐❝❛çã♦φ(x)_→xα_Dβ_φ₍_x₎ _{é ❝♦♥tí♥✉❛ ❡♠} _S_✳

❜✮ ❙❡ φ∈ S ❡♥tã♦ limh→0τhφ(x) = φ ❡♠ S✳

❝✮ ❙❡ φ∈ S ❡ h= (0, . . . , hi, . . . ,0)∈Rn ❡♥tã♦ ♦ s❡❣✉✐♥t❡ ❧✐♠✐t❡ ♦❝♦rr❡ ❡♠ S

φ₋τhφ

hi →

∂φ ∂xi

, q✉❛♥❞♦ _|h_{| →}0.

(17)

❞✮ S é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✳

❡✮ ❆ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ S ❡♠ S✳ ❢✮ ❖ ❡s♣❛ç♦ C∞

c (Rn) é ❞❡♥s♦ ❡♠ S✳

❣✮ S é ✉♠ ❡s♣❛ç♦ s❡♣❛rá✈❡❧✳

❖ ♣ró①✐♠♦ t❡♦r❡♠❛ ❡①♣r✐♠❡ ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ s✐♠♣❧❡s ♣❛r❛ ❛s ❞✐str✐❜✉✐çõ❡s t❡♠♣❡r❛✲ ❞❛s✳

❚❡♦r❡♠❛ ✶✳✶✳✾✳ ❯♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛rLs♦❜r❡_S é ✉♠❛ ❞✐str✐❜✉✐çã♦ t❡♠♣❡r❛❞❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C >0 ❡ ✐♥t❡✐r♦s m, ℓ t❛✐s q✉❡

|L(φ)| ≤C X

|α|≤ℓ,|β|≤m

pα,β(φ)

♣❛r❛ t♦❞❛ φ∈ S✱ ♦♥❞❡ pα,β(φ) = supx∈Rn

xα_Dβ_φ₍_x₎_✳

❉❡♠♦♥str❛çã♦✿ ❬❙❲❪

❚❡♦r❡♠❛ ✶✳✶✳✶✵✳ ❙❡u_{∈ S}′ _❡_φ_{∈ S} _{❡♥tã♦ ❛ ❝♦♥✈♦❧✉çã♦}_u_∗_φ_{❞❡✜♥❡ ✉♠❛ ❢✉♥çã♦ ❡♠}_C∞₍_Rn₎_✱

❞❛❞❛ ♣♦r f(x) = hu, τxφ˜i✱ ❧❡♥t❛♠❡♥t❡ ❝r❡s❝❡♥t❡ ❝♦♠ ❞❡r✐✈❛❞❛s ❧❡♥t❛♠❡♥t❡ ❝r❡s❝❡♥t❡s✳

❉❡♠♦♥str❛çã♦✿ ❬❙❲❪

❚❡♦r❡♠❛ ✶✳✶✳✶✶✳ ❙✉♣♦♥❤❛ q✉❡B :Lp₍_Rn₎_→_Lq₍_Rn₎_✱₁_≤_{p, q} _{≤ ∞}_{s❡❥❛ ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r}

❧✐♠✐t❛❞♦ ❡ q✉❡ ❝♦♠✉t❛ ❝♦♠ tr❛♥s❧❛çõ❡s✳ ❊♥tã♦✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ❞✐str✐❜✉✐çã♦ t❡♠♣❡r❛❞❛ u t❛❧ q✉❡ Bφ=u∗φ ♣❛r❛ t♦❞❛ φ∈ S✳

❉❡♠♦♥str❛çã♦✿ ❬❙❲❪

❉❡♥♦t❛♠♦s ♣♦r (Lp_{, L}q₎ _{♦ ❡s♣❛ç♦ ❞❛s ❞✐str✐❜✉✐çõ❡s t❡♠♣❡r❛❞❛s} _u _{∈ S}′ _{t❛✐s q✉❡ ❡①✐st❡}

✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛A t❛❧ q✉❡ ku∗φkLq₍Rn₎ ≤Akφk_Lp₍Rn₎ ♣❛r❛ t♦❞❛ φ∈ S✳ ❖ ❚❡♦r❡♠❛

✶✳✶✳✶✶✱ q✉❛♥❞♦p < _∞✱ ♠♦str❛ q✉❡ ❡①✐st❡ ✉♠❛ ❜✐❥❡çã♦ ❡♥tr❡ ♦ ❡s♣❛ç♦(Lp_{, L}q₎_{❡ ♦ ❡s♣❛ç♦ ❞♦s}

♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ❡ ❧✐♠✐t❛❞♦s ❞❡ Lp₍_Rn₎ _❡♠ _Lq₍_Rn₎ _{q✉❡ ❝♦♠✉t❛♠ ❝♦♠ tr❛♥s❧❛çõ❡s✳}

❚❡♦r❡♠❛ ✶✳✶✳✶✷✳ ❯♠❛ ❞✐str✐❜✉✐çã♦ u _∈ (L2_{, L}2₎ _{s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡} _b _∈ _L∞₍_Rn₎ _t❛❧

q✉❡ Fu = b✳ ◆❡st❡ ❝❛s♦✱ _kb_kL∞ é ❛ ♥♦r♠❛ ❞♦ ♦♣❡r❛❞♦r B : L2 ∩ S → L2 ❞❡✜♥✐❞♦ ♣♦r

Bφ=u∗φ✳ ❆❧é♠ ❞✐ss♦✱ F(u∗φ) = (Fu)(Fφ)✳

❉❡♠♦♥str❛çã♦✿ ❬❙❲❪

(18)

❚❡♦r❡♠❛ ✶✳✶✳✶✸✳ ❯♠❛ ❞✐str✐❜✉✐çã♦u_∈(L1_{, L}1₎_{s❡✱ ❡ s♦♠❡♥t❡ s❡✱} _u_{é ✉♠❛ ♠❡❞✐❞❛ ❞❡ ❇♦r❡❧}

✜♥✐t❛✳ ◆❡st❡ ❝❛s♦✱ ❛ ✈❛r✐❛çã♦ t♦t❛❧ ❞❛ ♠❡❞✐❞❛ u é ❛ ♥♦r♠❛ ❞♦ ♦♣❡r❛❞♦r B : L1_{∩ S →} _L1

❞❡✜♥✐❞♦ ♣♦r Bφ=u∗φ✳

❉❡♠♦♥str❛çã♦✿ ❬❙❲❪

❚❡♦r❡♠❛ ✶✳✶✳✶✹✳ ❙❡ 1 _≤ p, q _{≤ ∞}✱ 1/p+ 1/p′ _{= 1} _❡ ₁_/q _{+ 1}_/q′ _{= 1} _❡♥tã♦ ₍_Lp_{, L}q_{) =}

(Lq′

, Lp′

)✳

❉❡♠♦♥str❛çã♦✿ ❬❙❲❪

✶✳✷ ❋✉♥çã♦ ▼❛①✐♠❛❧ ❞❡ ❍❛r❞②✲▲✐tt❧❡✇♦♦❞

❉❡✜♥✐çã♦ ✶✳✷✳✶✳ ❙❡❥❛ f _∈L1

loc(Rn)✳ P❛r❛ x∈Rn ❞❡✜♥✐♠♦s

M f(x) = sup

r>0

1

|B(x, r)|

Z

B(x,r)|

f(y)_|dy. ✭✶✳✷✳✶✮

M f é ❞❡♥♦♠✐♥❛❞❛ ❢✉♥çã♦ ♠❛①✐♠❛❧ ❞❡ ❍❛r❞②✲▲✐tt❧❡✇♦♦❞✱ ❡ ♦ ♦♣❡r❛❞♦r M q✉❡ ❡♥✈✐❛ f ❡♠ M f é ❞❡♥♦♠✐♥❛❞♦ ♦♣❡r❛❞♦r ♠❛①✐♠❛❧ ❞❡ ❍❛r❞②✲▲✐tt❧❡✇♦♦❞✳

▲❡♠❛ ✶✳✷✳✶✳ ❬▲❡♠❛ ❞❡ ❈♦❜❡rt✉r❛❪ ❙❡❥❛ E _⊆Rn _{✉♠ s✉❜❝♦♥❥✉♥t♦ ♠❡♥s✉rá✈❡❧ ❡ s✉♣♦♥❤❛ q✉❡}

❡①✐st❛♠ ❜♦❧❛sBj ❝♦♠j = 1, . . . , N t❛✐s q✉❡E ⊆ B1∪. . .∪BN✳ ❊♥tã♦✱ ❡①✐st❡ ✉♠❛ s✉❜❝♦❧❡çã♦

{B′

j} t❛❧ q✉❡

✭✐✮ B′

j ∩Bk′ =∅ ♣❛r❛ t♦❞♦j 6=k❀

✭✐✐✮ ❊①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ c=c(n) q✉❡ ❞❡♣❡♥❞❡ s♦♠❡♥t❡ ❞❛ ❞✐♠❡♥sã♦ t❛❧ q✉❡

|E| ≤c

m

X

j=1

|B_j′|. ✭✶✳✷✳✷✮

❉❡♠♦♥str❛çã♦✿ ❘❡♦r❞❡♥❛♥❞♦ ❛ ❝♦❜❡rt✉r❛ ♣♦❞❡♠♦s s✉♣♦r q✉❡ ❛ ❝♦❧❡çã♦Bj =B(xj, rj), j =

1, . . . , N é t❛❧ q✉❡ r1 ≥r2 ≥. . .≥rN✳ ❙❡❥❛ B′1 =B1 ❡ ❞❡✜♥❛ ♦ ❝♦♥❥✉♥t♦

O1 ={Bj : Bj ∩B1′ =∅}.

❙❡ O1 =∅ ❡♥tã♦ Bj∩B1′ 6=∅ ♣❛r❛ t♦❞♦ j 6= 1✳ ◆❡st❡ ❝❛s♦✱ Bj ⊆ (B1′)∗

.

=B(x1,3r1)✱ ♣❛r❛

t♦❞♦j✳ ▲♦❣♦✱

E _{⊆ ∪}N

j=1Bj ⊆(B′1)∗

(19)

❞❡ ♦♥❞❡ ❝♦♥❝❧✉✐r✐❛♠♦s q✉❡

|E_{| ≤ |}B(x1,3r1)|= 3n|B1′|.

❙✉♣♦♥❤❛ q✉❡O1 6=∅✳ ◆❡st❡ ❝❛s♦✱ s❡❥❛ Bj0 t❛❧ q✉❡

❞✐❛♠Bj0 = max{❞✐❛♠Bj :Bj ∈ O1}

s❡❥❛ B′

2

.

=Bj0✳ ❉❡✜♥❛ ❛❣♦r❛ ♦ ❝♦♥❥✉♥t♦

O2 ={Bj ∈ O1 :Bj∩B2′ =∅}.

❙❡ O2 = ∅ ❡♥tã♦ ♣❛r❛ ❝❛❞❛ Bj ∈ O2 ♦❝♦rr❡r✐❛ q✉❡ Bj ∩B2′ 6= ∅✳ ◆❡st❡ ❝❛s♦ t♦❞♦ Bj ∈ O2

s❡r✐❛ t❛❧ q✉❡ Bj ⊆(B2′)∗✳ ▲♦❣♦✱

E ⊆ ∪N

j=1Bj ⊆(B1′)∗∪(B2′)∗

✐♠♣❧✐❝❛♥❞♦ q✉❡

|E| ≤3n(|B₁′|+|B₂′|).

❘❡♣❡t✐♥❞♦ ❡st❡ ❛r❣✉♠❡♥t♦ ♣♦❞❡r❡♠♦s ❡♥❝♦♥tr❛r ✉♠❛ s✉❜❝♦❧❡çã♦ {B′

1, . . . , Bm′ } ❡ ✉♠❛ s❡✲

q✉ê♥❝✐❛ ❞❡ ❝♦♥❥✉♥t♦sO1, . . . ,Om t❛✐s q✉❡

{Bj}Nj=1 = (∪mk=1{Bj ∈ Ok−1 : Bj ∩Bk′ 6=∅})∪ Om ✭✶✳✷✳✸✮

❡ ❝♦♠ Om =∅✳ ❆✜r♠❛♠♦s q✉❡

E _⊆(B′₁)∗ _∪. . ._∪(B_m′ )∗. ✭✶✳✷✳✹✮ P❛r❛ ♣r♦✈❛♠♦s ✭✶✳✷✳✹✮ é s✉✜❝✐❡♥t❡ ♠♦str❛r q✉❡Bj ⊂(B1′)∗∪. . .∪(Bm′ )∗✱ ♣❛r❛ ❝❛❞❛Bj ✜①❛❞♦✳

❙✉♣♦♥❤❛ q✉❡ Bj ♥ã♦ ♣❡rt❡♥ç❛ ❛ ❝♦❧❡çã♦ B1′, . . . , B′m ❝❛s♦ ❝♦♥trár✐♦✱ ❛ ❝♦♥❝❧✉sã♦ s❡r✐❛ ó❜✈✐❛✳

❙❡❣✉❡ ❞❡ ✭✶✳✷✳✸✮ q✉❡ Bj∩Bℓ′ 6=∅♣❛r❛ ❛❧❣✉♠ℓ = 1, . . . , m✱ ♠♦str❛♥❞♦ q✉❡ Bj ⊂(Bℓ′)∗ ♣❛r❛

❛❧❣✉♠ ℓ= 1, . . . , m ❝♦♥❝❧✉✐♥❞♦ ❛ ♣r♦✈❛ ❞❡ ✭✶✳✷✳✹✮✳ ●r❛ç❛s ❛ ✭✶✳✷✳✹✮ ✈❡♠♦s q✉❡

|E_|=

m

[

j=1

(B′

j)∗∩E

≤

m

X

j=1

|(B′

j)∗∩E| ≤ m

X

j=1

|(B′

j)∗|= 3n m

X

j=1

|B′

j|,

❝♦♥❝❧✉✐♥❞♦ ♣♦rt❛♥t♦ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛✳

▲❡♠❛ ✶✳✷✳✷✳ ❊①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ c=c(n)>0 q✉❡ só ❞❡♣❡♥❞❡ ❞❛ ❞✐♠❡♥sã♦✱ t❛❧ q✉❡

|{x_∈Rn : _|M f(x)_|> λ_{}| ≤} c(n)

λ kfkL1(Rn). ✭✶✳✷✳✺✮ ❉❡♠♦♥str❛çã♦✿ P❛r❛ ❝❛❞❛ λ >0 ✜①♦ ❞❡✜♥✐♠♦s ♦ ❝♦♥❥✉♥t♦ Eλ ={x∈ Rn : M f(x)> λ}✳

P♦r ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ |Eλ| = sup{|K| : K ⊂⊂ Eλ}✳ ❆ss✐♠ ♣❛r❛

❝❛❞❛ σ < _|Eλ| ✜①♦ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ K ⊂⊂ Eλ t❛❧ q✉❡ σ < |K| ≤ |Eλ|✳ ▲♦❣♦✱

♣❛r❛ t♦❞♦x_∈K✱ ❝♦♠♦M f(x)> λ✱ ❡①✐st❡ rx >0t❛❧ q✉❡

1

|B(x, rx)|

Z

B(x,rx)

|f(y)_|dy > λ. ✭✶✳✷✳✻✮

(20)

❈♦♠♦ K é ❝♦♠♣❛❝t♦ ❡①✐st❡ ✉♠❛ s✉❜❝♦❜❡rt✉r❛ ✜♥✐t❛ K _⊆B(x1, rx1)∪. . .∪B(xN, rxN)✳ ❆♣❧✐❝❛♥❞♦ ♦ ▲❡♠❛ ✶✳✷✳✶ ❛ ❡st❛ ❝♦❜❡rt✉r❛ ♣♦❞❡♠♦s ❡①✐❜✐r ✉♠❛ s✉❜❝♦❧❡çã♦ ❞❡ ❜♦❧❛s B′

1, . . . , Bm′ t❛✐s q✉❡

|K_{| ≤}c

m

X

j=1

|B_j′_|.

❆ss✐♠ ♣❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✷✳✻✮ t❡♠♦s q✉❡

|K_{| ≤}c

m

X

j=1

1

λ

Z

B′

j

|f(y)_|dy.

❙❡♥❞♦ ❛ ❝♦❧❡çã♦ ❞✐s❥✉♥t❛ ❝♦♥❝❧✉✐♠♦s q✉❡ |K_{| ≤} c λ

Z

∪m j=1Bj′

|f(y)_|dy.

▲♦❣♦✱ |K_{| ≤} _λc_kf_kL1₍_Rn₎✳ ❊♥tã♦ ❢❛③❡♥❞♦ σ → |E_λ| ♥❛ ❞❡s✐❣✉❛❧❞❛❞❡ σ < c

λkfkL1(Rn) ✈❡♠♦s

✜♥❛❧♠❡♥t❡ q✉❡ |Eλ| ≤ _λckfkL1₍_Rn₎✳ ❈♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳

❈♦r♦❧ár✐♦ ✶✳✷✳✶✳ ❙❡❥❛f ✉♠❛ ❢✉♥çã♦ ♠❡♥s✉rá✈❡❧ ❡ ❞❡✜♥✐❞❛ ❡♠Rn _{❡ s❡❥❛}_{λ >}₀_{✳ ❊♥tã♦ ✈❛❧❡}

❛ s❡❣✉✐♥t❡ ❡st✐♠❛t✐✈❛ ♣❛r❛ ❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❞♦ ❝♦♥❥✉♥t♦Eλ ={x∈Rn: M f(x)> λ}✿

|Eλ| ≤

c λ

Z

{x∈Rn_: _|_f₍_x₎_|_>λ/₂_}|

f(x)_|dx ✭✶✳✷✳✼✮

❉❡♠♦♥str❛çã♦✿ ❙❡❣✉❡ ❞♦ ▲❡♠❛ ✶✳✷✳✷ ❛♣❧✐❝❛❞♦ ❛ f1 =f χ{x:|f(x)|>λ/2}✳

Pr♦♣♦s✐çã♦ ✶✳✷✳✶✳ ❙❡ f =χB(0,1) ❡♥tã♦ M f /∈L1(Rn)✳

❉❡♠♦♥str❛çã♦✿ ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ x ∈ Rn _{t❡♠♦s q✉❡} _B₍₀_,₁₎ _⊆ _B₍_x,_|_x_|_{+ 1)}_{✳ ❆ss✐♠ ❛}

s❡❣✉✐♥t❡ ❡st✐♠❛t✐✈❛ ♦❝♦rr❡✿

M f(x) ≥ 1 |B(x,_|x_|+ 1)_|

Z

B(x,|x|+1)|

f(y)|dy

≥ 1

|B(x,|x|+ 1)|

Z

B(0,1)|

f(y)_|dy

= |B(0,1)|

|B(x,_|x_|+ 1)_| = 1

(_|x_|+ 1)n ≥

1 2n_|_x_|n.

▲♦❣♦✱ _Z

1≤|x|≤R

M f(x)dx_≥

Z

1≤|x|≤R

c_|x_|−ndx =c′lnR ✭✶✳✷✳✽✮

♠♦str❛♥❞♦ q✉❡ M f /∈L1₍_Rn₎_✳

❚❡♦r❡♠❛ ✶✳✷✳✶✳ ❙❡❥❛ f ∈L1₍_Rn₎ _{s✉♣♦rt❛❞❛ ♥✉♠❛ ❜♦❧❛} _B _⊂_Rn_{✳ ❊♥tã♦} _{M f} _∈_L1₍_B₎ _{s❡✱ ❡}

s♦♠❡♥t❡ s❡ _Z

B|

f(x)_|log+_|f(x)_|dx <_∞ ✭✶✳✷✳✾✮

(21)

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛B =B(x0, r)✳ ❙✉♣♦♥❤❛ q✉❡ ✭✶✳✷✳✾✮ ♦❝♦rr❛✱ ❡♥tã♦ Z

B

M f(x)dx=

Z _∞

0

|{x∈B : M f(x)> λ}|dλ

= 2

Z ∞

0 |{

x_∈B : M f(x)>2λ_}|dλ _≤2

Z 1

0 |

B_|dλ+

Z ∞

1 |

Eλ|dλ

≤2|B|+c

Z _∞

1

λ

Z

{x∈B: |f(x)|>λ}|

f(x)|dx dλ = 2|B|+c

Z

B|

f(x)|

Z _|f(x)|

1

λdλ

!

dx

= 2|B|+c

Z

B|

f(x)|log+|f(x)|dx <∞.

P♦r ♦✉tr♦ ❧❛❞♦✱ s✉♣♦♥❤❛ q✉❡R_BM f(x)dx <∞❡ ❞❡♥♦t❡B∗ ₌_B₍_x

0,2r)❡♥tã♦ R

B∗M f(x)dx <

∞✳ P❛r❛ ♣r♦✈❛r ❡st❡ ❢❛t♦ ♦❜s❡r✈❡ q✉❡

Z

B∗

M f(x)dx=

Z

B

M f(x)dx+

Z

B∗_\_B

M f(x)dx

❡ q✉❡ ❛❧é♠ ❞✐ss♦✱ ♣❛r❛ ♦ s✐♠étr✐❝♦x′ _❞❡_x_∈_B∗_\_B_{✱ ❝♦♠ r❡s♣❡✐t♦ à ❢r♦♥t❡✐r❛ ❞❛ ❜♦❧❛✱ t❡♠✲s❡}

B(x, r)⊂B(x′_,₃_r₎_{✳ ▲♦❣♦✱} 1

|B(x, r)|

Z

B(x,r)|

f(y)_|dy _≤ |B(x′,3r)| |B(x, r)|

1

|B(x′_,₃_r₎_|

Z

B(x′_,₃_r₎|

f(y)_|dy _≤c(n)M f(x′).

❖✉ s❡❥❛✱ M f(x)≤ cM f(x′₎ _{❞❡ ♦♥❞❡ ❝♦♥❝❧✉✐✲s❡ q✉❡} R

B∗_\_BM f(x)dx ≤ |

B∗_\_B_|

|B|

R

BM f(x)dx✳

❆ s❡❣✉✐r ♠♦str❛r❡♠♦s q✉❡ M f(x) _→ 0 q✉❛♥❞♦ _|x_{| → ∞}✳ ❉❡ ❢❛t♦✱ s❡❥❛ x _∈ Rn _{t❛❧ q✉❡}

|x−x0| ≥ 2r✳ P❛r❛ t♦❞♦ s ≤ |x−x0| −r é ❢❛t♦ q✉❡ B(x, s)∩B = ∅✳ ▲♦❣♦ s✉♣♦♥❤❛ q✉❡

s_{≥ |}x₋x0| −r ❡♥tã♦

1

|B(x, s)_|

Z

B(x,s)∩B(x0,r)

|f(t)|dt ≤ kfkL1(B) |B(x, s)_| ≤

kfkL1₍_B₎

(_|x₋x0| −r)n

.

P♦rt❛♥t♦✱ M f(x)≤ c

(|x−x0|−r)n✳ ▲♦❣♦ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡✱ ♣❛r❛ q✉❛❧q✉❡r t0 >0✱ Z

{x:M f(x)>t0}

M f(x)dx <_∞

P❛r❛ t0 = 1✱ Z

{x:M f(x)>1}

M f(x)dx_≥

Z _∞

1 |{

x: M f(x)> t_}|dt _≥

Z _∞

1

1 2n_t

Z

{x∈Rn_:_|_f₍_x₎_|_>t_}|

f(x)_|dx dt

= 1 2n

Z

Rn| f(x)|

Z _|f(x)|

1

dt t dx=

1 2n

Z

Rn|

f(x)| log+|f(x)|dx.

P♦rt❛♥t♦✱ _Z

Rn|

f(x)_| log+_|f(x)_|dx <_∞.

(22)

✶✳✸ ❚❡♦r❡♠❛ ❞❡ ■♥t❡r♣♦❧❛çã♦ ❞❡ ▼❛r❝✐♥❦✐❡✇✐❝③

❙❡❥❛♠ (X, µ) ❡ (Y, ν) ❡s♣❛ç♦s ❞❡ ♠❡❞✐❞❛✱ ❡ s❡❥❛ T ✉♠ ♦♣❡r❛❞♦r ❞❡ Lp₍_{X, µ}₎ _{à ✈❛❧♦r❡s ♥✉♠}

❡s♣❛ç♦ ❞❡ ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s ❞❡Y ❡♠ C✳ ❉✐③❡♠♦s q✉❡ T é ❞❡ t✐♣♦ ❢r❛❝♦ (p, q)✱ q <∞✱ s❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛C t❛❧ q✉❡

ν(_{y_∈Y : _|T f(y)_|> λ_})_≤

C_kf_kLp λ

q

,

♣❛r❛ t♦❞♦λ >0✳ ❉✐③❡♠♦s q✉❡ é ❞❡ t✐♣♦ ❢r❛❝♦(p,_∞)s❡ ❢♦r ✉♠ ♦♣❡r❛❞♦r ❧✐♠✐t❛❞♦ ❞❡Lp₍_{X, µ}₎

❡♠ L∞₍_{Y, ν}₎_{✳ ❉✐③❡♠♦s q✉❡} _T _{é ❢♦rt❡} ₍_{p, q}₎ _{s❡ ❢♦r ❧✐♠✐t❛❞♦ ❞❡} _Lp₍_{X, µ}₎ _❡♠ _Lq₍_{Y, ν}₎_✳

❆ ❞❡s✐❣✉❛❧❞❛❞❡ t✐♣♦ ❢r❛❝❛ (p, p)é ✐♠♣♦rt❛♥t❡ ♣♦rq✉❡ é ❝❤❛✈❡ ♣❛r❛ ♣r♦✈❛r q✉❡ ♦ ♦♣❡r❛❞♦r

é ❧✐♠✐t❛❞♦ t✐♣♦ ❢♦rt❡ (p, p) ✈✐❛ ♦ t❡♦r❡♠❛ ❞❡ ✐♥t❡r♣♦❧❛çã♦ ❞❡ ▼❛r❝✐♥❦✐❡✇✐❝③✳

▲❡♠❛ ✶✳✸✳✶✳ ❙❡ T é ❢♦rt❡ (p, q) ❡♥tã♦ é ❢r❛❝♦ (p, q)✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛λ >0 ✜①♦ ❡ ❞❡✜♥❛ Eλ ={y∈Y : |T f(y)|> λ}✱ ❡♥tã♦

ν(Eλ) =

Z

Eλ

1dν ≤

Z

Eλ

T f_λ(x)

q

dν ≤ kT fk

q Lq λq ≤

CkfkLp λ

q

✭✶✳✸✳✶✮

❆ r❡❧❛çã♦ ❡♥tr❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❢r❛❝❛ (p, q)❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡♠ q✉❛s❡ t♦❞❛ ❛ ♣❛rt❡ é ❞❛❞❛

♣❡❧♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦

❚❡♦r❡♠❛ ✶✳✸✳✶✳ ❙❡❥❛ {Tǫ} ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ❞❡✜♥✐❞♦s s♦❜r❡ Lp(X, µ) ❡

❞❡✜♥❛ ♦ ♦♣❡r❛❞♦r ♠❛①✐♠❛❧

T∗f(x) = sup

ǫ>0 |Tǫf(x)|.

❙❡ T∗ _{é ❢r❛❝♦} ₍_{p, q}₎ _{❡♥tã♦ ♦ ❝♦♥❥✉♥t♦}

F =nf _∈Lp(X, µ) : lim

ǫ→0Tǫf(x) = f(x) q✳t✳♣✳ o

é ❢❡❝❤❛❞♦ ❡♠ Lp₍_{X, µ}₎_✳

❉❡♠♦♥str❛çã♦✿ ❉❛❞♦ f ∈ F✱ s❡❥❛ {fn} ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❢✉♥çõ❡s ❞❡ F q✉❡ ❝♦♥✈❡r❣❡

❛ f ❡♠ Lp_{✳ ▲♦❣♦✱ ♣❛r❛ ❝❛❞❛} _n _∈ _N _{✜①♦ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♠❡❞✐❞❛ ♥✉❧❛} _E

n t❛❧ q✉❡

limǫ→0Tǫfn(x) = fn(x) ♣❛r❛ t♦❞♦ x ∈ X \En✳ P❛r❛ ♠♦str❛r q✉❡ f ∈ F✱ ♣r♦✈❛r❡♠♦s q✉❡

❞❛❞♦ q✉❛❧q✉❡r λ >0 ♦ s❡❣✉✐♥t❡ ❢❛t♦ ♦❝♦rr❡

µ(_{x_∈X : lim sup

ǫ−→0 |

Tǫf(x)−f(x)|> λ}) = 0 ✭✶✳✸✳✷✮

❈♦♠ ❡ss❡ ♦❜❥❡t✐✈♦✱ ❝♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦

Ω(f, x)= lim sup.

ǫ−→0 |

Tǫf(x)−f(x)|.

(23)

➱ ❢á❝✐❧ ✈❡r q✉❡ Ω(g, x) = 0 ♣❛r❛ t♦❞♦ ❡❧❡♠❡♥t♦ g _∈F ❡♥tã♦

Ω(f, x) _≤ Ω(f ₋fn, x) + Ω(fn, x)

= lim sup

ǫ−→0 |

Tǫ(f −fn)(x)−(f −fn)(x)| ≤T∗(f−fn)(x) +|(f −fn)(x)|

❆ss✐♠✱

µ({x∈X : Ω(f, x)> λ}) ≤ µ({x∈X : T∗(f−fn)(x)> λ/2})

+µ({x∈X : |(f−fn)(x)|> λ/2}).

❈♦♠♦ T∗ _{é ❞❡ t✐♣♦ ❢r❛❝♦} ₍_{p, q}₎ _{❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛}_C _{t❛❧ q✉❡}

µ(_{x_∈X : T∗(f ₋fn)(x)> λ/2}| ≤

2C

λ kf −fnkp

q

. ✭✶✳✸✳✸✮

❆❧é♠ ❞✐ss♦✱ s❡❣✉❡ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❚❝❤❡❜②❝❤❡✈ ❛ s❡❣✉✐♥t❡ ❡st✐♠❛t✐✈❛

µ({x∈X : |(f−fn)(x)|> λ/2})≤

2

λkf−fnkp

p

. ✭✶✳✸✳✹✮

❉❡st❡ ♠♦❞♦✱ ❢❛③❡♥❞♦ n_{→ ∞} ❡♠ ✭✶✳✸✳✸✮ ❡ ✭✶✳✸✳✹✮ t❡♠♦s q✉❡ µ(_{x_∈X : Ω(f, x)> λ_}) = 0

♣❛r❛ ❝❛❞❛ λ >0✜①♦✳ ❉❡s❞❡ q✉❡✱

µ({x∈X : Ω(f, x)>0})≤

∞

X

k=1

µ({x∈X : Ω(f, x)>1/k}) = 0,

s❡❣✉❡ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛

❙❡❥❛(X, µ)✉♠ ❡s♣❛ç♦ ❞❡ ♠❡❞✐❞❛ ❡ s❡❥❛ f :X →C✉♠❛ ❢✉♥çã♦ ♠❡♥s✉rá✈❡❧✳ ❈❤❛♠❛♠♦s

❛ ❢✉♥çã♦ ωf : (0,∞)→[0,∞]✱ ❞❛❞❛ ♣♦r

ωf(λ) =µ({x∈X : |f(x)|> λ}),

❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❞❡ f ❛ss♦❝✐❛❞❛ ❛ ♠❡❞✐❞❛ µ✳

Pr♦♣♦s✐çã♦ ✶✳✸✳✶✳ ❙❡❥❛ φ : [0,_∞] _→ [0,_∞] ✉♠❛ ❢✉♥çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧✱ ❝r❡s❝❡♥t❡ ❡ t❛❧ q✉❡

φ(0) = 0✳ ❊♥tã♦✱ _Z

X

φ(_|f(x)_|)dµ(x) =

Z _∞

0

φ′(λ)ωf(λ)dλ. ✭✶✳✸✳✺✮

❉❡♠♦♥str❛çã♦✿ ❙❡❣✉❡ ❞♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦ q✉❡

Z

X

φ(_|f(x)_|)dµ(x) =

Z

X

Z _|f(x)|

0

φ′(λ)dλ

!

dµ(x) =

Z _∞

0 Z

{x∈X:|f(x)|>λ}

φ′(λ)dµ(x)

dλ

=

Z ∞

0

φ′(λ)µ(_{x_∈X : _|f(x)_|> λ_})dλ=

Z ∞

0

φ′(λ)ωf(λ)dλ.

(24)

❆♣❧✐❝❛♥❞♦ ❛ Pr♦♣♦s✐çã♦ ✶✳✸✳✶ à ❢✉♥çã♦ φ(λ) =λp_✱ _φ′₍_λ_{) =} _pλp−1 _{♦❜t❡♠♦s}

kf_kp_Lp =

Z

X |

f(x)_|pdµ(x) = p

Z _∞

0

λp−1ωf(λ)dλ. ✭✶✳✸✳✻✮

❉❡✜♥✐çã♦ ✶✳✸✳✶✳ ❯♠ ♦♣❡r❛❞♦r T ❞❡✜♥✐❞♦ ♥♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❛s ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s é ❞✐t♦ s✉❜❧✐♥❡❛r s❡

✭✐✮ |T(f1+f2)(x)| ≤ |T f1(x)|+|T f2(x)|

✭✐✐✮ |T(αf)(x)_{| ≤ |}α_||T f(x)_|, _∀α_∈C✳

Pr♦♣♦s✐çã♦ ✶✳✸✳✷✳ ❙❡❥❛♠ 0< p1 < p < p2 <∞✱ ❡♥tã♦✱ Lp(Rn)⊂Lp1(Rn) +Lp2(Rn).

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛f _∈Lp₍_Rn₎ _{❡ ❞❡✜♥✐♠♦s} _E ₌_{_x_∈_Rn_:_|_f₍_x₎_|_>₁_}_❡ _Ec ₌_{_x_∈_Rn_:

|f(x)| ≤1}✳ ❊♥tã♦✱

f(x) =f(x)χE(x) +f(x)(1−χE(x)) =f1(x) +f2(x)∈Lp1(Rn) +Lp2(Rn).

❚❡♦r❡♠❛ ✶✳✸✳✷ ✭■♥t❡r♣♦❧❛çã♦ ❞❡ ▼❛r❝✐♥❦✐❡✇✐❝③✮✳ ❙❡❥❛♠ (X, µ) ❡ (Y, ν) ❡s♣❛ç♦s ♠❡♥s✉rá✲

✈❡✐s✱ 1_≤p1 < p2 ≤ ∞✱ ❡ s❡❥❛ T ✉♠ ♦♣❡r❛❞♦r s✉❜❧✐♥❡❛r ❞❡ Lp1(X, µ) +Lp2(X, µ) ❛♦ ❡s♣❛ç♦

❞❡ ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s ❡♠ Y q✉❡ é ❢r❛❝♦ (p1, p1) ❡ ❢r❛❝♦ (p2, p2)✳ ❊♥tã♦ T é ❢♦rt❡ (p, p)

♣❛r❛ p1 < p < p2✳

❉❡♠♦♥str❛çã♦✿ ❉❛❞♦ f _∈ Lp_{✱ ♣❛r❛ ❝❛❞❛} _{λ >} ₀ _{✜①♦ ❞❡❝♦♠♣♦♠♦s} _f _❝♦♠♦ _f

1 +f2✱ ♦♥❞❡

f1(x) =f(x)χ{x:|f(x)|>cλ}(x)✱ ❡ f2(x) =f(x)χ{x:|f(x)|≤cλ}(x)s❡♥❞♦ c >0✉♠❛ ❝♦♥st❛♥t❡ ❛ s❡r

✜①❛❞❛✳ ➱ ❢á❝✐❧ ✈❡r q✉❡ f1 ∈Lp1(µ)❡ f2 ∈Lp2(µ)❀ ❛❧é♠ ❞✐ss♦✱

|T f(x)| ≤ |T f1(x)|+|T f2(x)|,

❛ss✐♠

µ(_{x_∈X : _|T f(x)_|> λ_})_≤µ(_{x_∈X : _|T f1(x)|> λ/2}) +µ({x∈X : |T f2(x)|> λ/2}),

♦✉ s❡❥❛✱

ωT(f)(λ)≤ωT(f1)(λ/2) +ωT(f2)(λ/2).

❈♦♥s✐❞❡r❡✲s❡ ❞♦✐s ❝❛s♦s✿

❈❛s♦ ✶✿ p2 = ∞✳ ❊s❝♦❧❤❛ c = ₂_A1₂✱ ♦♥❞❡ A2 é t❛❧ q✉❡ kT gk∞ ≤ A2kgk∞✳ ♣❛r❛ t♦❞❛

g ∈ L∞_{✳ ◆❡st❡ ❝❛s♦ ♠♦str❛r❡♠♦s q✉❡} _ω

T f2(λ/2) = 0✳ ❉❡ ❢❛t♦✱ s❡❣✉❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ f2

(25)

q✉❡ kf2k∞ ≤ cλ = 2Aλ2✳ ❆ss✐♠✱ kT f2k∞ ≤ A2kf2k∞ ≤

A2λ 2A2 =

λ

2✳ ❉❡st❡ ♠♦❞♦✱ ❡①✐st❡ ✉♠

❝♦♥❥✉♥t♦ ♠❡♥s✉rá✈❡❧E t❛❧ q✉❡ µ(E) = 0 ❡ _|T f2(x)|< λ₂ ♣❛r❛ t♦❞♦ x∈X\E✳ ◆♦t❡ q✉❡

ωT f2(λ/2) =µ({x∈X : |T f2(x)|> λ/2})

≤µ(_{x_∈E : _|T f2(x)|> λ/2}) +µ({x∈X\E : |T f2(x)|> λ/2})

≤µ(E) +µ(_{x_∈X_\E : _|T f2(x)|> λ/2}).

❆✜r♠❛♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ {x _∈ X_\E : _|T f2(x)| > λ/2} = ∅ ♣♦✐s ❝❛s♦ ❡①✐st❛ x ∈ X\E

t❛❧ q✉❡ |T f2(x)| > λ/2 ❡♥tã♦ λ/2 < |T f2(x)| ≤ kT f2k∞ ≤ λ/2 ♦ q✉❡ ❣❡r❛ ✉♠ ❛❜s✉r❞♦ ❡

♣♦rt❛♥t♦ ωT f2(λ/2) = 0✳

P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❢r❛❝❛ (p1, p1)

ωT f1(λ/2)≤

2A1

λ kf1kLp1

p1

;

♣♦rt❛♥t♦✱

kT f_kp_Lp ≤p

Z ∞

0

λp−1_ω

T f1(λ/2)dλ

≤p

Z ∞

0

λp−1−p1₍₂_A 1)p1

Z

X|

f1(x)|p1dµ dλ

=p(2A1)p1 Z ∞

0

λp−1−p1 Z

{x:|f(x)|>cλ}|

f(x)_|p1_{dµ dλ}

=p(2A1)p1 Z

X|

f(x)_|p1 Z ∞

0

λp−1−p1_χ

{x:|f(x)|>cλ}dλ dµ

=p(2A1)p1 Z

X|

f(x)_|p1

Z _|f(x)|/c

0

λp−1−p1_{dλ dµ}₌ p

p−p1

(2A1)p1(2A2)p−p1kfkp_Lp.

❈❛s♦ ✷✿ p2 <∞✳ ❆❣♦r❛ t❡♠♦s

ωT fi(λ/2)≤

2Ai

λ kfikLpi

pi

, i= 1,2.

◆♦t❡ q✉❡

kT f_kp_p _≤p

Z _∞

0

λp−1ωT f(λ)dλ

≤p

Z _∞

0

λp−1ωT f1(λ/2)dλ+p Z _∞

0

λp−1ωT f2(λ/2)dλ

(26)

❉❡st♦ ♦❜t❡♠♦s q✉❡

kT f_kp_Lp ≤ p

Z ∞

0

λp−1−p1₍₂_A 1)p1

Z

X |

f1(x)|p1dµ dλ

+p

Z ∞

0

λp−1−p2₍₂_A 2)p2

Z

X|

f2(x)|p2dµ dλ

≤ p(2A1)p1 Z ∞

0

λp−1−p1 Z

{x:|f(x)|>cλ}|

f(x)_|p1_{dµ dλ}

+p(2A2)p2 Z ∞

0

λp−1−p2 Z

{x:|f(x)|≤cλ}|

f(x)_|p2_{dµ dλ}

= p(2A1)p1 Z

X|

f(x)_|p1 Z ∞

0

λp−1−p1_χ

{x:|f(x)|>cλ}dλ dµ

+p(2A2)p2 Z

X|

f(x)_|p2 Z ∞

0

λp−1−p2_χ

{x:|f(x)|≤cλ}dλ dµ

= p(2A1)p1 Z

X|

f(x)_|p1

Z _|f(x)|/c

0

λp−1−p1_{dλ dµ}

+p(2A2)p2 Z

X|

f(x)_|p2 Z _∞

|f(x)|/c

λp−1−p2_{dλ dµ}

=

p(2A1)p1cp1−p

p−p1

+p(2A2)

p2cp2−p

p2−p

kf_kp_Lp.

▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱

kT fkp ≤2p1/p

1

p−p1

+ 1

p2−p 1/p

A11−θAθ2kfkp

♦♥❞❡

1

p = θ p2

+1−θ

p1

, 0< θ <1.

❚❡♦r❡♠❛ ✶✳✸✳✸✳ ❖ ♦♣❡r❛❞♦r M é ❢r❛❝♦ (1,1) ❡ ❢♦rt❡ (p, p)✱ 1< p_{≤ ∞}✳

❉❡♠♦♥str❛çã♦✿ P❡❧♦ ▲❡♠❛ ✶✳✷✳✷ t❡♠♦s q✉❡ ♦ ♦♣❡r❛❞♦r M é ❢r❛❝♦ (1,1) ❡ é ✐♠❡❞✐❛t♦ ❞❛

❞❡✜♥✐çã♦ q✉❡ kM f_kL∞ ≤ kfk_L∞✱ ❛ss✐♠ ♣❡❧♦ t❡♦r❡♠❛ ❞❡ ✐♥t❡r♣♦❧❛çã♦ ❞❡ ▼❛r❝✐♥❦✐❡✇✐❝③

t❡♠♦s q✉❡M é ❢♦rt❡ (p, p)✱ ❝♦♠ 1< p_{≤ ∞}✳

✶✳✹ ❚❡♦r❡♠❛ ❞❡ ❉✐❢❡r❡♥❝✐❛çã♦ ❞❡ ❘✐❡♠❛♥♥✲▲❡❜❡s❣✉❡

▲❡♠❛ ✶✳✹✳✶✳ ❙❡❥❛ f _∈C0₍_Rn₎ _{❡♥tã♦✱}

lim

r→0

1

|B(x, r)|

Z

B(x,r)

f(t)dt=f(x).

(27)

❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ f é ❝♦♥tí♥✉❛ ❡♠ x✱ ❡♥tã♦ ♣❛r❛ t♦❞♦ ǫ > 0✱ ❡①✐st❡ δ > 0 t❛❧ q✉❡

|t₋x_|< δ ✐♠♣❧✐❝❛ _|f(t)₋f(x)_|< ǫ✳ ❚♦♠❛♥❞♦ r < δ t❡♠♦s B(x, r)_⊂B(x, δ)✳ ▲♦❣♦✱

_|_B₍_{x, r}1 ₎_| Z

B(x,r)

f(t)dt₋f(x)

=

_|_B₍_{x, r}1 ₎_| Z

B(x,r)

(f(t)₋f(x))dt

≤ 1

|B(x, r)_|

Z

B(x,r)|

f(t)₋f(x)_|dt < ǫ.

❚❡♦r❡♠❛ ✶✳✹✳✶ ✭❉✐❢❡r❡♥❝✐❛çã♦ ❞❡ ▲❡❜❡s❣✉❡✮✳ ❙❡ f _∈Lp₍_Rn₎_{✱ ❝♦♠} ₁_≤_p_{≤ ∞}_{✱ ❡♥tã♦}

lim

r→0

1

|B(x, r)_|

Z

B(x,r)

f(t)dt =f(x)

♣❛r❛ q✉❛s❡ t♦❞♦ x_∈Rn_✳

❉❡♠♦♥str❛çã♦✿ Pr♦✈❛r❡♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡ ♦ ❧✐♠✐t❡

lim

r→0

1

|B(x, r)_|

Z

B(x,r)

f(t)dt

❡①✐st❡ ♣❛r❛ t♦❞♦ x ♥♦ ❝♦♠♣❧❡♠❡♥t❛r ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♠❡❞✐❞❛ ♥✉❧❛✳ ❉❡ ❢❛t♦✱ s❡❥❛ f ✉♠❛ ❢✉♥çã♦ ♠❡♥s✉rá✈❡❧ ❡ ❞❡✜♥❛ ♣❛r❛ t♦❞♦ x_∈Rn _{❛ ❛♣❧✐❝❛çã♦}

Ω(f, x) = lim sup

r→0

1

|B(x, r)_|

Z

B(x,r)

f(t)dt−lim inf

r→0

1

|B(x, r)_|

Z

B(x,r)

f(t)dt.

◆♦t❡ q✉❡ ❛ Ω(f, x)≥0✱ é s✉❜❛❞✐t✐✈❛ ❡ s❛t✐s❢❛③ ❛ s❡❣✉✐♥t❡ ❞❡s✐❣✉❛❧❞❛❞❡ Ω(f, x)≤2M f(x). ❙✉♣♦♥❞♦ q✉❡ f _∈ Lp₍_Rn₎ _{♠♦str❛r❡♠♦s q✉❡} _Ω(_{f, x}_{) = 0} _{♣❛r❛ q✉❛s❡ t♦❞♦} _x _∈ _Rn_{✱ ✐✳❡✳✱}

|{x ∈ Rn _{: Ω(}_{f, x}₎ _>₀_}| _{= 0} _{♣❛r❛ q✉❛s❡ t♦❞♦ ♣♦♥t♦} _x _∈ _Rn_{✱ ❞❡ ♦♥❞❡ ♣♦❞❡r❡♠♦s ❝♦♥❝❧✉✐r}

♥♦ss❛ ❛✜r♠❛çã♦✳

➱ s✉✜❝✐❡♥t❡ ♠♦str❛r q✉❡ ♣❛r❛ t♦❞♦ a > 0✱ ♦ ❝♦♥❥✉♥t♦ {x ∈ Rn _{: Ω(}_{f, x}₎ _{> a}_} _t❡♠

♠❡❞✐❞❛ ♥✉❧❛✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ C0

c(Rn) é ❞❡♥s♦ ❡♠ Lp ♣❛r❛ t♦❞♦ 1 ≤ p ≤ ∞✱ ❞❛❞♦ ǫ > 0

❡①✐st❡ g _∈C0

c(Rn) t❛❧ q✉❡ kf −gkLp < ǫ✳ ❙❡❣✉❡ ❞❛ s✉❜❛❞✐t✐✈✐❞❛❞❡ ❞❛ ❢✉♥çã♦ Ω ❛ s❡❣✉✐♥t❡ ❞❡s✐❣✉❛❧❞❛❞❡

Ω(f, x) = Ω(f ₋g +g, x)_≤Ω(f ₋g, x) + Ω(g, x).

❈♦♠♦ g é ❝♦♥tí♥✉❛ ♣❡❧♦ ▲❡♠❛ ✶✳✹✳✶ t❡♠♦s q✉❡ Ω(g, x) = 0 ♣❛r❛ t♦❞♦ x _∈ Rn_{✳ ❆ss✐♠✱}

Ω(f, x)≤Ω(f −g, x)❞❡ ♦♥❞❡ ❝♦♥❝❧✉í♠♦s q✉❡

|{x_∈Rn: Ω(f, x)> a_{}| ≤ |{}x_∈Rn :M(f ₋g, x)> a/2_}|. ✭✶✳✹✳✶✮ ❊♠ s❡❣✉✐❞❛✱ ❞✐✈✐❞✐r❡♠♦s ❛ ❞❡♠♦♥str❛çã♦ ❡♠ ❞♦✐s ❝❛s♦s

❈❛s♦ ✶✿ p >1✳

(28)

◆❡st❡ ❝❛s♦✱ ♣❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞♦ ♦♣❡r❛❞♦r ♠❛①✐♠❛❧ ❞❡ ❍❛r❞②✲▲✐tt❧❡✇♦♦❞ ♥♦ ❡s♣❛ç♦ Lp

♣❛r❛ 1 < p _{≤ ∞} ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ C t❛❧ q✉❡ _kM(f ₋g)_kLp ≤ Ckf −gk_Lp✳ ❉❡❝♦rr❡ ❞❡ ✭✶✳✹✳✶✮ ❡ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❚❝❤❡❜✐❝❤❡✈ ❛ s❡❣✉✐♥t❡ ❡st✐♠❛t✐✈❛

|{x_∈Rn : Ω(f, x)> a_{}| ≤ |{}x_∈Rn : M(f₋g)(x)> a/2_}|

≤ ₍_a/1₂₎_pkM(f₋g)_kp_Lp ≤

2p_C

ap kf −gk p Lp ≤

2p_Cǫp

ap .

❈❛s♦ ✷✿ p= 1

◆❡st❡ ❝❛s♦✱ ✉s❛r❡♠♦s ♦ ▲❡♠❛ ✶✳✷✳✷ ♥❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✹✳✶✮ ♣❛r❛ ♦❜t❡r♠♦s ❛ s❡❣✉✐♥t❡ ❡st✐♠❛t✐✈❛

|{x_∈Rn _{: Ω(}_{f, x}₎_{> a}

}| ≤ |{x_∈Rn_:_M₍_f

−g)(x)> a/2_}|

≤ 2C

a kf −gkL1 ≤

2Cǫ a .

❊♠ ❛♠❜♦s ♦s ❝❛s♦s✱ ❝♦♠♦ǫ♣♦❞❡ s❡r ❡s❝♦❧❤✐❞♦ ❞❡ ❢♦r♠❛ ❛r❜✐trár✐❛ ♣r♦✈❛♠♦s q✉❡✱ ♣❛r❛ t♦❞♦ a >0✱|{x∈Rn _{: Ω(}_{f, x}₎_{> a}_}|_{= 0}_{✳ ❊♠ ♣❛rt✐❝✉❧❛r t❡♠♦s ❛ s❡❣✉✐♥t❡ ❝♦♥s❡q✉ê♥❝✐❛}

|{x_∈Rn _{: Ω(}_{f, x}₎_>₀

}| = _{| ∪}∞_k₌₁_{x_∈Rn _{: Ω(}_{f, x}₎_>₁_/k

}|

≤

∞

X

k=1

|{x∈Rn _{: Ω(}_{f, x}₎_>₁_/k_}|_{= 0}

❞❡ ♦♥❞❡✱ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ♠❡♥s✉rá✈❡❧ E ❝♦♠ ♠❡❞✐❞❛ ♥✉❧❛ t❛❧ q✉❡Ω(f, x) = 0 ♣❛r❛ t♦❞♦

x_∈Rn_\_E✳

❈♦♥s✐❞❡r❡♠♦s ❞❛q✉✐ ♣♦r ❞✐❛♥t❡ ❛ ❢✉♥çã♦ ϕ(x) = 1

|B(0,1)_|χB(0,1)(x) ❡ s❡❥❛ ϕr(x)

.

= 1

rnϕ(x/r)✳ ◆♦t❛♥❞♦ q✉❡ χB(0,1)(y/r) =χB(0,r)(y) t❡♠♦s q✉❡

ϕr(y) =

1

rnϕ(y/r) =

1

rn

1

|B(0,1)_|χB(0,1)(y/r) = 1

|B(0, r)_|χB(0,r)(y).

▲♦❣♦✱

1

|B(x, r)|

Z

B(x,r)

f(t)dt = 1

|B(0, r)|

Z

B(0,r)

f(x₋y)dy

=

Z

Rn

f(x₋y) 1

|B(0, r)_|χB(0, r)(y)dy= (f∗ϕr)(x).

❈♦♠♦f∗ϕr →f ❡♠ Lp q✉❛♥❞♦ r→0t❡♠♦s q✉❡ ❡①✐st❡F ♠❡♥s✉rá✈❡❧✱ ❝♦♠|F|= 0 t❛❧ q✉❡

f _∗ϕr(x)→f(x)♣❛r❛ t♦❞♦ x∈Rn\F✳ ▲♦❣♦ ♣❛r❛ t♦❞♦x∈Rn\ {E∪F} t❡♠♦s

lim

r→0

1

|B(x, r)|

Z

B(x,r)

f(t)dt=f(x).

❈♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳