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Start Over Creator Urciuolo, Marta

2. On the $H^{p}$-$L^{q}$ boundedness of some fractional integral operators

Creator:
Rocha, Pablo and Urciuolo, Marta
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
integral operator and Hardy space
Language:
English
Description:
Let $A_{1},\dots ,A_{m}$ be $n\times n$ real matrices such that for each $1\leq i\leq m,$ $A_{i}$ is invertible and $A_{i}-A_{j}$ is invertible for $i\neq j$. In this paper we study integral operators of the form $$ Tf( x) =\int k_{1}( x-A_{1}y) k_{2}( x-A_{2}y) \dots k_{m}( x-A_{m}y) f( y) {\rm d} y, $$ $k_{i}( y) =\sum _{j\in \mathbb Z}2^{jn/{q_{i}}}\varphi _{i,j}( 2^{j}y) $, $1\leq q_{i}<\infty,$ $1/{q_{1}}+1/{q_{2}}+\dots +1/{q_{m}}=1-r,$ $0\leq r<1,$ and $\varphi _{i,j}$ satisfying suitable regularity conditions. We obtain the boundedness of $T\colon H^{p}( \mathbb {R} ^{n}) \rightarrow L^{q} (\mathbb {R}^{n}) $ for $ 0<p<1/{r}$ and $1/{q}=1/{p}-r.$ We also show that we can not expect the $H^{p}$-$H^{q}$ boundedness of this kind of operators.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

3. The type set for some measures on $\mathbb R^{2n}$ with $n$-dimensional support

Creator:
Ferreyra, E., Godoy, T., and Urciuolo, Marta
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
singular measures and convolution operators
Language:
English
Description:
Let $\varphi _1,\dots ,\varphi _n$ be real homogeneous functions in $C^\infty (\mathbb R^n-\lbrace 0\rbrace )$ of degree $k\ge 2$, let $\varphi (x) =(\varphi _1(x),\dots ,\varphi _n(x))$ and let $\mu $ be the Borel measure on $\mathbb R^{2n}$ given by \[ \mu (E) =\int _{\mathbb R^n}\chi _E(x,\varphi (x))\, |x|^{\gamma -n}\mathrm{d}x \] where $\mathrm{d}x$ denotes the Lebesgue measure on $\mathbb R^n$ and $\gamma >0$. Let $T_\mu $ be the convolution operator $T_\mu f(x)=(\mu *f)(x)$ and let \[ E_\mu =\lbrace (1/p,1/q)\:\Vert T_\mu \Vert _{p,q}<\infty ,\hspace{5.0pt}1\le p, \,q\le \infty \rbrace . \] Assume that, for $x\ne 0$, the following two conditions hold: $\det ({\mathrm d}^2\varphi (x) h)$ vanishes only at $h=0$ and $\det ({\mathrm d} \varphi (x)) \ne 0$. In this paper we show that if $\gamma >n(k+1)/3$ then $E_\mu $ is the empty set and if $\gamma \le n(k+1)/3$ then $E_\mu $ is the closed segment with endpoints $D=\bigl (1-\frac{\gamma }{n(k+1)},1-\frac{2\gamma }{n(k+1)}\bigr )$ and $D^{\prime }=\bigl (\frac{2\gamma }{n(1+k)},\frac{\gamma }{n(1+k)}\bigr )$. Also, we give some examples.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

4. Weighted inequalities for integral operators with some homogeneous kernels

Creator:
Riveros, María Silvina and Urciuolo, Marta
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
weights and integral operators
Language:
English
Description:
In this paper we study integral operators of the form \[ Tf(x)=\int | x-a_1y|^{-\alpha _1}\dots | x-a_my|^{-\alpha _m}f(y)\mathrm{d}y, \] $\alpha _1+\dots +\alpha _m=n$. We obtain the $L^p(w)$ boundedness for them, and a weighted $(1,1)$ inequality for weights $w$ in $A_p$ satisfying that there exists $c\ge 1$ such that $w( a_ix) \le cw( x)$ for a.e. $x\in \mathbb R^n$, $1\le i\le m$. Moreover, we prove $\Vert Tf\Vert _{{\mathrm BMO}}\le c\Vert f\Vert _\infty $ for a wide family of functions $f\in L^\infty ( \mathbb R^n)$.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public