Let \lbrace T_t\rbrace_{t>0} be the semigroup of linear operators generated by a Schrödinger operator –A = \Delta –V where V is nonnegative potential that belongs to certain reverse Hölder class. We define Hardy space H^1_A means maximal function associated with . Atomic and Riesz transforms characterizations are shown.

In this work we extend the theory of classical Hardy space $$H^1$$ to rational Dunkl setting. Specifically, let $$\Delta $$ be Laplacian on a Euclidean $$\mathbb {R}^N$$ . On half-space {R}_+\times \mathbb , consider systems conjugate $$(\partial _t^2+\Delta _{\mathbf {x}})$$ -harmonic functions satisfying an appropriate uniform $$L^1$$ condition. We prove that boundary values such harmonic functions, which constitute real $$H^1_{\Delta }$$ can characterized in several different ways, namely...

Let $A=-{\mit \Delta } +V$ be a Schrödinger operator on ${{\mathbb R}}^d$, $d\geq 3$, where $V$ is nonnegative potential satisfying the reverse Hölder inequality with an exponent $q>d/2$. We say that $f$ element of $H^p_A$ if maximal function

where w is a weight from the Muckenhoupt class A 2 , V nonnegative potential that belongs to certain reverse Hölder with respect measure w(x) dx, andLet {Tt} t>0 be semigroup of linear operators generated by -L.We say function f an element space H 1 L if maximal operator Mf (x) = sup |Ttf (x)| (R d (w(x) dx)).A special atomic decomposition proved.

This paper is perhaps the first attempt at a study of Hardy space $$H^1$$ in rational Dunkl setting. Following Uchiyama's approach, we characterize atomically and by means heat maximal operator. We also obtain Fourier multiplier theorem for . These results are proved here one-dimensional case product case.

Let {K t } t>0 be the semigroup of linear operators generated by a Schrödinger operator −L = Δ − V (x) on ℝ d , ≥ 3, where 0 satisfies −1 ∈ L ∞. We say that an 1-function f belongs to Hardy space ${H^{1}_{L}}$ if maximal function ℳ sup |K (x)| 1 (ℝ ). prove (−Δ)1/2 −1/2 is isomorphism with classical H 1(ℝ ) whose inverse 1/2(−Δ)−1/2. As corollary we obtain characterized Riesz transforms $R_{j}=\frac {\partial }{\partial x_{j}}L^{-1\slash 2}$ .

Let $$T_t=e^{-tL}$$ be a semigroup of self-adjoint linear operators acting on $$L^2(X,\mu )$$ , where $$(X,d,\mu is space homogeneous type. We assume that $$T_t$$ has an integral kernel $$T_t(x,y)$$ which satisfies the upper and lower Gaussian bounds: $$\begin{aligned} \frac{C_1}{\mu (B(x,\sqrt{t}))} \exp \left( {-\,c_1d(x,y)^2/ t} \right) \le T_t(x,y)\le \frac{C_2}{\mu {-\,c_2 d(x,y)^2/ . \end{aligned}$$ By definition, f belongs to $$H^1(L)$$ if $$\Vert f\Vert _{{H^1(L)}}=\Vert \sup...

Let $\varDelta$ be the Dunkl Laplacian on $\mathbb R^N$ associated with a normalized root system $R$ and multiplicity function $k(\alpha)\geq 0$. We say that $f$ belongs to Hardy space $H^1_{\varDelta}$ if nontangential maximal functi

Let {Tt}t>0 be the semigroup generated by a Schrodinger operator −A = ∆−V , where V is nonnegative polynomial on R. We say that f in H A associated with if maximal function Mf(x) supt>0 |Ttf(x)| belongs to L(R). characterize elements of space for 0 < p ≤ 1 special atomic decomposition.

Abstract It is well known that the compactly supported wavelets cannot belong to class . This also true for with exponential decay. We show one can construct in are “almost” of decay and, moreover, they band-limited. do this by showing we adapt construction Lemarié-Meyer [LM] found [BSW] so obtain band-limited, C ∞ -wavelets on R have subexponential decay, is, every 0 &lt; ε 1, there exits &gt; such Moreover, all its derivatives The proof constructive and uses Gevrey classes functions.

Let $\{ K_t\} _{t>0}$ be the semigroup of linear operators generated by a Schrödinger operator $-L={\mit \Delta } -V$ with $V\geq 0$. We say that $f$ belongs to $H_L^1$ if $\| \mathop {\rm sup}_{t>0}|K_tf(x)|\, \| _{L^1(dx)}<\infty $. state conditions

It is proved that Triebel-Lizorkin spaces for some Laguerre and Hermite expansions are well-defined.

For a Schrödinger operator A = -Δ + V, where V is nonnegative polynomial, we define Hardy $H_A^1$ space associated with A. An atomic characterization of shown.