We prove that spectral projections of Laplace-Beltrami operator on the m-complex unit sphere E∆ S2m−1 ([0, R)) are uniformly bounded as an operator from H(S) to L(S) for all p ∈ (1,∞). We also show that the Bochner-Riesz conjecture is true when restricted to cylindrically symmetric functions on R × R. Suppose that L is a positive definite, self-adjoint operator acting on L(X,μ), where X is a me...

We utilize the Bellman function technique to prove a bilinear dimension-free inequality for the Hermite operator. The Bellman technique is applied here to a non-local operator, which at first did not seem to be possible. An indispensable tool in order to make the proofs dimension-free is a certain linear algebra lemma concerning three bilinear forms. As a consequence of our bilinear inequality ...

Let {X1, . . . ,X2n,T} be a real basis for the Heisenberg Lie algebra hn. In this paper, we calculate the explicit kernels of the Riesz transforms Rj = XjL 1 2 , j = 1, 2, · · · , 2n on the Heisenberg group Hn. Here L = − ∑2n j=1 X 2 j is the sub-Laplacian operator. We show that ∫∞ −∞Rjdt = Rj for j = 1, 2, · · · , 2n where Rj ’s are the regular Riesz transforms on R . We also construct the “mi...

Let X and Y be compact Hausdorff spaces, and E, F be Banach lattices. Let C(X,E) denote the Banach lattice of all continuous E-valued functions on X equipped with the pointwise ordering and the sup norm. We prove that if there exists a Riesz isomorphism Φ : C(X,E) → C(Y, F ) such that Φf is non-vanishing on Y if and only if f is non-vanishing on X, then X is homeomorphic to Y , and E is Riesz i...

In this paper, we study approximate duals of $g$-frames and fusion frames in Hilbert $C^ast-$modules. We get some relations between approximate duals of $g$-frames and biorthogonal Bessel sequences, and using these relations, some results for approximate duals of modular Riesz bases and fusion frames are obtained. Moreover, we generalize the concept of $Q-$approximate duality of $g$-frames and ...

The Dirac operators Ly = i 1 0 0 −1 dy dx + v(x)y, y = y1 y2 , x ∈ [0, π], with L 2-potentials v(x) = 0 P (x) Q(x) 0 , P,Q ∈ L 2 ([0, π]), considered on [0, π] with periodic, antiperiodic or Dirichlet boundary conditions (bc), have discrete spectra, and the Riesz projections SN = 1 2πi |z|=N− 1 2 (z − L bc) −1 dz, Pn = 1 2πi |z−n|= 1 2 (z − L bc) −1 dz are well-defined for |n| ≥ N if N is suffi...

In this paper we use the lifting scheme to construct biorthogonal spline wavelet bases on regularly refined non-uniform grids. The wavelets have at least one vanishing moment and on each resolution level they form an L2 Riesz basis. Furthermore we are interested in determining the exact range of Sobolev exponents for which the complete multilevel system forms a Riesz basis. Hereto we need to ex...

Let L = −div(A(x)∇) be a second order elliptic operator with real, symmetric, bounded measurable coefficients on Rn or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed p > 2, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform ∇(L)−1/2 on the Lp space. As an application, for 1 < p < 3+ ε, we establish the Lp boundedness...

In his studies of mixing conditions on Markov chains, Rosenblatt [32; 33, Chap. 71 used the Riesz convexity (interpolation) theorem to compare different measures of dependence between two given families of random variables on a probability space. Rosenblatt [34] also suggested that by using other results in operator theory, one might be able to obtain more information about the relationships be...

The Dual-Phase-Lagging (DPL) equation is formulated as an abstract differential equation. In the absence of a heat source term the DPL equation with homogeneous boundary conditions generates a contraction semigroup. The exact expression of the semigroup is achieved. It is proved that the associated eigenfunctions form a Riesz basis. The stability of semigroup is proved. Moreover, it is also sho...