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Register a new userHigher Order Riesz Transforms for Laguerre Expansions
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In this paper we investigate Lp-boundedness properties for the higher order Riesz transforms associated with Laguerre operators. Also we prove that the k-th Riesz transform is a principal value singular integral operator (modulus a constant times of the function when k is even). To establish our results we exploit a new identity connecting Riesz transforms in the Hermite and Laguerre settings.
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The Riesz transform is a natural multi-dimensional extension of the Hilbert transform, and it has been the object of study for many years due to its nice mathematical properties. More recently, the Riesz transform and its variants have been used to construct complex wavelets and steerable wavelet frames in higher dimensions. The flip side of this approach, however, is that the Riesz transform of a wavelet often has slow decay. One can nevertheless overcome this problem by requiring the original wavelet to have sufficient smoothness, decay, and vanishing moments. In this paper, we derive necessary conditions in terms of these three properties that guarantee the decay of the Riesz transform and its variants, and as an application, we show how the decay of the popular Simoncelli wavelets can be improved by appropriately modifying their Fourier transforms. By applying the Riesz transform to these new wavelets, we obtain steerable frames with rapid decay.
In the work of S. Petermichl, S. Treil and A. Volberg it was explicitly constructed that the Riesz transforms in any dimension $n geq 2$ can be obtained as an average of dyadic Haar shifts provided that an integral is nonzero. It was shown in the paper that when $n=2$, the integral is indeed nonzero (negative) but for $n geq 3$ the nonzero property remains unsolved. In this paper we show that the integral is nonzero (negative) for $n=3$. The novelty in our proof is the delicate decompositions of the integral for which we can either find their closed forms or prove an upper bound.
Let $p(cdot): mathbb R^nto(0,infty)$ be a variable exponent function satisfying that there exists a constant $p_0in(0,p_-)$, where $p_-:=mathop{mathrm {ess,inf}}_{xin mathbb R^n}p(x)$, such that the Hardy-Littlewood maximal operator is bounded on the variable exponent Lebesgue space $L^{p(cdot)/p_0}(mathbb R^n)$. In this article, via investigating relations between boundary valued of harmonic functions on the upper half space and elements of variable exponent Hardy spaces $H^{p(cdot)}(mathbb R^n)$ introduced by E. Nakai and Y. Sawano and, independently, by D. Cruz-Uribe and L.-A. D. Wang, the authors characterize $H^{p(cdot)}(mathbb R^n)$ via the first order Riesz transforms when $p_-in (frac{n-1}n,infty)$, and via compositions of all the first order Riesz transforms when $p_-in(0,frac{n-1}n)$.
A pair of dual frames with almost exponentially localized elements (needlets) are constructed on $RR_+^d$ based on Laguerre functions. It is shown that the Triebel-Lizorkin and Besov spaces induced by Laguerre expansions can be characterized in terms of respective sequence spaces that involve the needlet coefficients.
In this paper we consider $L^p$ boundedness of some commutators of Riesz transforms associated to Schr{o}dinger operator $P=-Delta+V(x)$ on $mathbb{R}^n, ngeq 3$. We assume that $V(x)$ is non-zero, nonnegative, and belongs to $B_q$ for some $q geq n/2$. Let $T_1=(-Delta+V)^{-1}V, T_2=(-Delta+V)^{-1/2}V^{1/2}$ and $T_3=(-Delta+V)^{-1/2} abla$. We obtain that $[b,T_j] (j=1,2,3)$ are bounded operators on $L^p(mathbb{R}^n)$ when $p$ ranges in a interval, where $b in mathbf{BMO}(mathbb{R}^n)$. Note that the kernel of $T_j (j=1,2,3)$ has no smoothness.
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