We prove L2 x L2 to weak L1 estimates for some novel bilinear maximal operators of Kakeya and lac... more We prove L2 x L2 to weak L1 estimates for some novel bilinear maximal operators of Kakeya and lacunary type thus extending to this setting, the works of Cordoba and of Nagel, Stein and Wainger.
In this article, we conduct a study of integral operators defined in terms of nonconvolution type... more In this article, we conduct a study of integral operators defined in terms of nonconvolution type kernels with singularities of various degrees. The operators that fall within our scope of research include fractional integrals, fractional derivatives, pseudodifferential operators, Calderón-Zygmund operators, and many others. The main results of this article are built around the notion of an operator calculus that connects operators with different kernel singularities via vanishing moment conditions and composition with fractional derivative operators. We also provide several boundedness results on weighted and unweighted distribution spaces, including homogeneous Sobolev, Besov, and Triebel-Lizorkin spaces, that are necessary and sufficient for the operator's vanishing moment properties, as well as certain behaviors for the operator under composition with fractional derivative and integral operators. As applications, we prove T 1 type theorems for singular integral operators with different singularities, boundedness results for pseudodifferential operators belonging to the forbidden class S 0 1,1 , fractional order and hyper-singular paraproduct boundedness, a smooth-oscillating decomposition for singular integrals, sparse domination estimates that quantify regularity and oscillation, and several operator calculus results. It is of particular interest that many of these results do not require L 2-boundedness of the operator, and furthermore, we apply our results to some operators that are known not to be L 2-bounded.
We obtain boundedness for the bilinear spherical maximal function in a range of exponents that in... more We obtain boundedness for the bilinear spherical maximal function in a range of exponents that includes the Banach triangle and a range of L p with p < 1. We also obtain counterexamples that are asymptotically optimal with our positive results on certain indices as the dimension tends to infinity.
In this work we study boundedness of Littlewood-Paley-Stein square functions associated to multil... more In this work we study boundedness of Littlewood-Paley-Stein square functions associated to multilinear operators. We prove weighted Lebesgue space bounds for square functions under relaxed regularity and cancellation conditions that are independent of weights, which is a new result even in the linear case. For a class of multilinear convolution operators, we prove necessary and sufficient conditions for weighted Lebesgue space bounds. Using extrapolation theory, we extend weighted bounds in the multilinear setting for Lebesgue spaces with index smaller than one.
Proceedings of the American Mathematical Society, 2010
Let B δ be the class of all h × δh rectangles in the plane with h > 0 and 0 < δ < 1 2. The orient... more Let B δ be the class of all h × δh rectangles in the plane with h > 0 and 0 < δ < 1 2. The orientation of the rectangles is arbitrary. Form the maximal operator GM f (x) = sup 0<δ< 1 2 sup x∈R∈B δ 1 | log δ| • |R| R |f (y)| dy. Note the logarithmic term in the average. It is shown that GM is a bounded maximal operator in L 2 (R 2). The case of a fixed δ is due to Córdoba.
Annales Academiae Scientiarum Fennicae Mathematica, 2013
In the present work we extend a local Tb theorem for square functions of Christ [3] and Hofmann [... more In the present work we extend a local Tb theorem for square functions of Christ [3] and Hofmann [17] to the multilinear setting. We also present a new BM O type interpolation result for square functions associated to multilinear operators. These square function bounds are applied to prove a multilinear local Tb theorem for singular integral operators.
We prove L2 x L2 to weak L1 estimates for some novel bilinear maximal operators of Kakeya and lac... more We prove L2 x L2 to weak L1 estimates for some novel bilinear maximal operators of Kakeya and lacunary type thus extending to this setting, the works of Cordoba and of Nagel, Stein and Wainger.
In this article, we conduct a study of integral operators defined in terms of nonconvolution type... more In this article, we conduct a study of integral operators defined in terms of nonconvolution type kernels with singularities of various degrees. The operators that fall within our scope of research include fractional integrals, fractional derivatives, pseudodifferential operators, Calderón-Zygmund operators, and many others. The main results of this article are built around the notion of an operator calculus that connects operators with different kernel singularities via vanishing moment conditions and composition with fractional derivative operators. We also provide several boundedness results on weighted and unweighted distribution spaces, including homogeneous Sobolev, Besov, and Triebel-Lizorkin spaces, that are necessary and sufficient for the operator's vanishing moment properties, as well as certain behaviors for the operator under composition with fractional derivative and integral operators. As applications, we prove T 1 type theorems for singular integral operators with different singularities, boundedness results for pseudodifferential operators belonging to the forbidden class S 0 1,1 , fractional order and hyper-singular paraproduct boundedness, a smooth-oscillating decomposition for singular integrals, sparse domination estimates that quantify regularity and oscillation, and several operator calculus results. It is of particular interest that many of these results do not require L 2-boundedness of the operator, and furthermore, we apply our results to some operators that are known not to be L 2-bounded.
We obtain boundedness for the bilinear spherical maximal function in a range of exponents that in... more We obtain boundedness for the bilinear spherical maximal function in a range of exponents that includes the Banach triangle and a range of L p with p < 1. We also obtain counterexamples that are asymptotically optimal with our positive results on certain indices as the dimension tends to infinity.
In this work we study boundedness of Littlewood-Paley-Stein square functions associated to multil... more In this work we study boundedness of Littlewood-Paley-Stein square functions associated to multilinear operators. We prove weighted Lebesgue space bounds for square functions under relaxed regularity and cancellation conditions that are independent of weights, which is a new result even in the linear case. For a class of multilinear convolution operators, we prove necessary and sufficient conditions for weighted Lebesgue space bounds. Using extrapolation theory, we extend weighted bounds in the multilinear setting for Lebesgue spaces with index smaller than one.
Proceedings of the American Mathematical Society, 2010
Let B δ be the class of all h × δh rectangles in the plane with h > 0 and 0 < δ < 1 2. The orient... more Let B δ be the class of all h × δh rectangles in the plane with h > 0 and 0 < δ < 1 2. The orientation of the rectangles is arbitrary. Form the maximal operator GM f (x) = sup 0<δ< 1 2 sup x∈R∈B δ 1 | log δ| • |R| R |f (y)| dy. Note the logarithmic term in the average. It is shown that GM is a bounded maximal operator in L 2 (R 2). The case of a fixed δ is due to Córdoba.
Annales Academiae Scientiarum Fennicae Mathematica, 2013
In the present work we extend a local Tb theorem for square functions of Christ [3] and Hofmann [... more In the present work we extend a local Tb theorem for square functions of Christ [3] and Hofmann [17] to the multilinear setting. We also present a new BM O type interpolation result for square functions associated to multilinear operators. These square function bounds are applied to prove a multilinear local Tb theorem for singular integral operators.
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