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### About the Lp-Boundedness of Integral Operators with Kernels of the Form K1 (x-y)K2(x+y).

Mathematica Scandinavica

### About Certain Singular Kernels K(x,y) = K1(x-y)K2(x+y).

Mathematica Scandinavica

### Endpoint bounds for convolution operators with singular measures

Colloquium Mathematicae

Let $S\subset {ℝ}^{n+1}$ be the graph of the function $\varphi :{\left[-1,1\right]}^{n}\to ℝ$ defined by $\varphi \left({x}_{1},\cdots ,{x}_{n}\right)={\sum }_{j=1}^{n}{|{x}_{j}|}^{{\beta }_{j}},$ with 1<${\beta }_{1}\le \cdots \le {\beta }_{n},$ and let $\mu$ the measure on ${ℝ}^{n+1}$ induced by the Euclidean area measure on S. In this paper we characterize the set of pairs (p,q) such that the convolution operator with $\mu$ is ${L}^{p}$-${L}^{q}$ bounded.

### On some singular integral operatorsclose to the Hilbert transform

Colloquium Mathematicae

Let m: ℝ → ℝ be a function of bounded variation. We prove the ${L}^{p}\left(ℝ\right)$-boundedness, 1 < p < ∞, of the one-dimensional integral operator defined by ${T}_{m}f\left(x\right)=p.v.\int k\left(x-y\right)m\left(x+y\right)f\left(y\right)dy$ where $k\left(x\right)={\sum }_{j\in ℤ}{2}^{j}{\phi }_{j}\left({2}^{j}x\right)$ for a family of functions ${{\phi }_{j}}_{j\in ℤ}$ satisfying conditions (1.1)-(1.3) given below.

### Convolution operators with anisotropically homogeneous measures on ${ℝ}^{2n}$ with n-dimensional support

Colloquium Mathematicae

Let ${\alpha }_{i},{\beta }_{i}>0$, 1 ≤ i ≤ n, and for t > 0 and x = (x₁,...,xₙ) ∈ ℝⁿ, let $t•x=\left({t}^{\alpha ₁}x₁,...,{t}^{\alpha ₙ}xₙ\right)$, $t\circ x=\left({t}^{\beta ₁}x₁,...,{t}^{\beta ₙ}xₙ\right)$ and $||x||={\sum }_{i=1}^{n}{|{x}_{i}|}^{1/{\alpha }_{i}}$. Let φ₁,...,φₙ be real functions in ${C}^{\infty }\left(ℝⁿ-0\right)$ such that φ = (φ₁,..., φₙ) satisfies φ(t • x) = t ∘ φ(x). Let γ > 0 and let μ be the Borel measure on ${ℝ}^{2n}$ given by $\mu \left(E\right)={\int }_{ℝⁿ}{\chi }_{E}{\left(x,\phi \left(x\right)\right)||x||}^{\gamma -\alpha }dx$, where $\alpha ={\sum }_{i=1}^{n}{\alpha }_{i}$ and dx denotes the Lebesgue measure on ℝⁿ. Let ${T}_{\mu }f=\mu \ast f$ and let $||{T}_{\mu }{||}_{p,q}$ be the operator norm of ${T}_{\mu }$ from ${L}^{p}\left({ℝ}^{2n}\right)$ into ${L}^{q}\left({ℝ}^{2n}\right)$, where the ${L}^{p}$ spaces are taken with respect to the Lebesgue measure. The type set ${E}_{\mu }$ is defined by ${E}_{\mu }=\left(1/p,1/q\right):||{T}_{\mu }{||}_{p,q}<\infty ,1\le p,q\le \infty$. In the case ${\alpha }_{i}\ne {\beta }_{k}$ for 1 ≤ i,k ≤ n we characterize the type set under...

### Restriction theorems for the Fourier transform to homogeneous polynomial surfaces in ℝ³

Studia Mathematica

Let φ:ℝ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let Σ = (x,φ(x)): |x| ≤ 1 and let σ be the Borel measure on Σ defined by $\sigma \left(A\right)={\int }_{B}{\chi }_{A}\left(x,\phi \left(x\right)\right)dx$ where B is the unit open ball in ℝ² and dx denotes the Lebesgue measure on ℝ². We show that the composition of the Fourier transform in ℝ³ followed by restriction to Σ defines a bounded operator from ${L}^{p}\left(ℝ³\right)$ to ${L}^{q}\left(\Sigma ,d\sigma \right)$ for certain p,q. For m ≥ 6 the results are sharp except for some border points.

### Convolution operators with homogeneous singular measures on ${ℝ}^{3}$ of polynomial type. The remainder case.

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

### Fourier restriction estimates to mixed homogeneous surfaces.

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

### ${L}^{p}$-improving properties for measures on ${ℝ}^{4}$ supported on homogeneous surfaces in some non elliptic cases.

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

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