Weighted boundedness of Marcinkiewicz integral operators and related singular integral operators
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Yanping Chen
Abstract
When α ≥ 0 and Ω Lq(Sn−1) (q > 1), μΩ,α and τΩ,α can be defined by μΩ,α(f )(x) = (∫R2−2tα FΩ,t(f )(x)2dt) ½ and τΩ,α(f )(x) = ∫R 2−tαFΩ,t(f )(x)dt, where FΩ,t(f )(x) = 2−t 2t ∫0 b(s) ∫ Sn−1 Ω(y′)f (x − sy′)dσ(y′)ds. This paper gives the boundedness of μΩ,α and τΩ,α from the homogeneous weighted Sobolev space Lpα (w) to Lp(w), where p, q, and w satisfy certain conditions. Furthermore, this paper shows that the Marcinkiewicz integral operators μΩ,α,S and μ∗ Ω,α,λ are also bounded from Lpα (w) to Lp(w).
Abstract
When α ≥ 0 and Ω Lq(Sn−1) (q > 1), μΩ,α and τΩ,α can be defined by μΩ,α(f )(x) = (∫R2−2tα FΩ,t(f )(x)2dt) ½ and τΩ,α(f )(x) = ∫R 2−tαFΩ,t(f )(x)dt, where FΩ,t(f )(x) = 2−t 2t ∫0 b(s) ∫ Sn−1 Ω(y′)f (x − sy′)dσ(y′)ds. This paper gives the boundedness of μΩ,α and τΩ,α from the homogeneous weighted Sobolev space Lpα (w) to Lp(w), where p, q, and w satisfy certain conditions. Furthermore, this paper shows that the Marcinkiewicz integral operators μΩ,α,S and μ∗ Ω,α,λ are also bounded from Lpα (w) to Lp(w).
Chapters in this book
- Frontmatter I
- Foreword V
- Contents VII
- Hardy spaces associated to the Kohn Laplacian on a family of model domains in Cn+1 1
- Affine mixed Rayleigh quotients 29
- Riesz potentials and Hardy–Morrey spaces 49
- A conjecture regarding compact Lie groups and its consequences 79
- The singular Fredholm integral operators and related integral equations of Chandrasekhar type 89
- Extended multichannel weighted Schatten p-norm minimization for color image denoising 105
- The boundedness of a bilinear oscillatory integral 123
- On sharp Sobolev-type inequalities for multidimensional Cauchy measures 135
- Banach lattice-valued q-variation and convexity 153
- Weighted boundedness of Marcinkiewicz integral operators and related singular integral operators 167
- A spectral comparison note 195
- On existence and nonexistence of nonnegative solutions to heat equation on Riemannian manifolds 207
- The Neumann problem for the Laplacian in half-spaces 219
- An inequality on the mass of image of rectifiable chain under Lipschitz map 253
- Boundedness of general alternative Gaussian singular integrals on Gaussian variable Lebesgue spaces 261
- Remarks on weak–strong uniqueness for two-fluid model 281
- Coercive inequalities on Carnot groups and applications 291
- On exponential Yang–Mills fields and p-Yang–Mills fields 317
- Index 359
Chapters in this book
- Frontmatter I
- Foreword V
- Contents VII
- Hardy spaces associated to the Kohn Laplacian on a family of model domains in Cn+1 1
- Affine mixed Rayleigh quotients 29
- Riesz potentials and Hardy–Morrey spaces 49
- A conjecture regarding compact Lie groups and its consequences 79
- The singular Fredholm integral operators and related integral equations of Chandrasekhar type 89
- Extended multichannel weighted Schatten p-norm minimization for color image denoising 105
- The boundedness of a bilinear oscillatory integral 123
- On sharp Sobolev-type inequalities for multidimensional Cauchy measures 135
- Banach lattice-valued q-variation and convexity 153
- Weighted boundedness of Marcinkiewicz integral operators and related singular integral operators 167
- A spectral comparison note 195
- On existence and nonexistence of nonnegative solutions to heat equation on Riemannian manifolds 207
- The Neumann problem for the Laplacian in half-spaces 219
- An inequality on the mass of image of rectifiable chain under Lipschitz map 253
- Boundedness of general alternative Gaussian singular integrals on Gaussian variable Lebesgue spaces 261
- Remarks on weak–strong uniqueness for two-fluid model 281
- Coercive inequalities on Carnot groups and applications 291
- On exponential Yang–Mills fields and p-Yang–Mills fields 317
- Index 359
