Poisson semigroups and singular integrals

1986/05/01 by Björn E. J. Dahlberg · 6 citations
Mathematics · Computer Science · #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations

paper · pdf · doi:10.1090/s0002-9939-1986-0831384-0

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D subset-of bold upper R Superscript n"> <mml:semantics> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo> ⊂ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">D ⊂ \mathbf Rⁿ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a Lipschitz domain and consider the bilinear form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript upper D Endscripts u left-parenthesis partial-differential v slash partial-differential y right-parenthesis d upper P"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mo> ∫ </mml:mo> <mml:mi>D</mml:mi> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>u</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mi>v</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mi>y</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> <mml:mi>P</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">∫ _D u ( ∂ v/∂ y )dP</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We show that the form is bounded if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v"> <mml:semantics> <mml:mi>v</mml:mi> <mml:annotation encoding="application/x-tex">v</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is harmonic with boundary values in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">L²</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u"> <mml:semantics> <mml:mi>u</mml:mi> <mml:annotation encoding="application/x-tex">u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is smooth with its nontangential maximal function in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">L²</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript upper D Endscripts dist StartSet upper P comma partial-differential upper D EndSet StartAbsoluteValue grad u EndAbsoluteValue squared d upper P greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mo> ∫ </mml:mo> <mml:mi>D</mml:mi> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>dist</mml:mtext> </mml:mrow> <mml:mrow> <mml:mo></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>P</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mi>D</mml:mi> </mml:mrow> <mml:mo></mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>grad </mml:mtext> </mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mi>d</mml:mi> <mml:mi>P</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">∫ _D \text dist \ P,∂ D \ | \text grad u |²dP &gt; ∞ </mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

Cited by

• Regularity and Neumann problems for operators with real coefficients satisfying Carleson conditions
• The L p Dirichlet problem for second order elliptic operators and a p-adapted square function
• L² estimates for commutators of the Dirichlet-to-Neumann Map associated to elliptic operators with complex-valued bounded measurable coefficients on ℝⁿ⁺¹₊
• Commutator Estimates for the Dirichlet-to-Neumann Map of Stokes Systems in Lipschitz Domains
• Functional calculus on weighted Sobolev spaces for the Laplacian on rough domains
• BMO solvability and the A_∞ condition for elliptic operators

Related

• A relation between pointwise convergence of functions and convergence of functionals
• On the derivative of a polynomial
• On a theorem of Scott and Swarup
• Trudinger type inequalities in 𝐑^𝐍 and their best exponents
• An explicit density estimate for Dirichlet 𝐿-series