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Register a new userSharp weighted bounds for fractional integral operators
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The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights is investigated. Sharp boundsare obtained for both the fractional integral operators and the associated fractional maximal functions. As an application improved Sobolev inequalities are obtained. Some of the techniques used include a sharp off-diagonal version of the extrapolation theorem of Rubio de Francia and characterizations of two-weight norm inequalities.
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In this note we prove the estimate $M^{sharp}_{0,s}(Tf)(x) le c,M_gamma f(x)$ for general fractional type operators $T$, where $M^{sharp}_{0,s}$ is the local sharp maximal function and $M_gamma$ the fractional maximal function, as well as a local version of this estimate. This allows us to express the local weighted control of $Tf$ by $M_gamma f$. Similar estimates hold for $T$ replaced by fractional type operators with kernels satisfying H{o}rmander-type conditions or integral operators with homogeneous kernels, and $M_gamma $ replaced by an appropriate maximal function $M_T$. We also prove two-weight, $L^p_v$-$L^q_w$ estimates for the fractional type operators described above for $1<p< q<infty$ and a range of $q$. The local nature of the estimates leads to results involving generalized Orlicz-Campanato and Orlicz-Morrey spaces.
We prove in this note one weight norm inequalities for some positive Bergman-type operators.
Volterra integral operators with non-sign-definite degenerate kernels $A(x,t)= sum_{k=0}^n A_k(x,t)$, $A_k(x,t)= a_k (x) t^k$, are studied acting from one weighted $L_2$ space on $(0,+infty)$ to another. Imposing an integral doubling condition on one of the weights, it is shown that the operator with the kernel $A(x,t)$ is bounded if and only $n+1$ operators with kernels $A_k(x,t)$ are all bounded. We apply this result to describe spaces of pointwise multipliers in weighted Sobolev spaces on $(0,+infty)$.
In this work we focus on substantial fractional integral and differential operators which play an important role in modeling anomalous diffusion. We introduce a new generalized substantial fractional integral. Generalizations of fractional substantial derivatives are also introduced both in Riemann-Liouville and Caputo sense. Furthermore, we analyze fundamental properties of these operators. Finally, we consider a class of generalized substantial fractional differential equations and discuss the existence, uniqueness and continuous dependence of solutions on initial data.
For 1<p<infty and for weight w in A_p, we show that the r-variation of the Fourier sums of any function in L^p(w) is finite a.e. for r larger than a finite constant depending on w and p. The fact that the variation exponent depends on w is necessary. This strengthens previous work of Hunt-Young and is a weighted extension of a variational Carleson theorem of Oberlin-Seeger-Tao-Thiele-Wright. The proof uses weighted adaptation of phase plane analysis and a weighted extension of a variational inequality of Lepingle.
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