Mathematics

In this paper, we study the L^p-Bochner-Riesz mean summability problem related to the spectrum of the Sturm-Liouville operator in L^p([a, b], \omega). Our purpose is to establish suitable conditions under which the Bochner-Riesz expansion of a function f \in L^p([a, b], \omega) converges to f in L^p([a, b], \omega). Then we apply this result in the case of Jacobi polynomials and two generalisations of Slepian's basis.

We develop a biparameter theory for matrix weights and provide various biparameter matrix-weighted bounds for Journé operators as well as other central operators under the assumption of the product matrix Muckenhoupt condition. In particular, we provide a complete theory for biparameter Journé operator bounds on matrix-weighted L 2 spaces. We also achieve bounds in the general case of matrix-weighted L p spaces, for 1<p<??for paraproduct-free Journé operators. We use two different methods: direct estimates using matrix-weighted square functions and a sparse domination technique in terms of square functions. Finally, we expose an open problem involving a matrix-weighted Fefferman Stein inequality, on which our methods rely in the general setting of matrix-weighted bounds for arbitrary Journé operators and p??.

We prove L p estimates for various multi-parameter bi- and trilinear operators with symbols acting on fibers of the two-dimensional functions. In particular, this yields estimates for the general bi-parameter form of the twisted paraproduct studied in arXiv:1011.6140.

Simple upper and lower bounds are established for the integral ??x 0 e ?�βt t ν L ν (t)dt , where x>0 , ν>?? , 0<β<1 and L ν (x) is the modified Struve function of the first kind. These bounds complement and improve on existing results, through either sharper bounds or increased ranges of validity. In deriving our bounds, we obtain some monotonicity results and inequalities for products of the modified Struve function of the first kind and the modified Bessel function of the second kind K ν (x) , as well as a new bound for the ratio L ν (x)/ L ν?? (x) .

Simple upper and lower bounds are obtained for the integral ∫ x 0 e −γt t ν I ν (t)dt , x>0 , ν>− 1 2 , 0<γ<1 . Most of our bounds for this integral are tight as x→∞ . We apply one of our inequalities to bound some expressions involving this integral. Two of these expressions appear in Stein's method for variance-gamma approximation, and our bounds will allow for a technical advancement to be made to the method.

In this paper, we define Bézier variant of generalized Bernstein-Durrmeyer type operators of second order, introduced by Ana et al. Then, we find an error estimate in terms of terms of Ditzian Totik modulus of smoothness. Next, we study the rate of approximation for a larger class of functions of bounded variation.

In this paper, we prove the existence of a nonnegative parameter-dependent (nonlinear) C 2 ( R 2 ) extension operator with bounded depth.

In this paper we settle the two-dimensional case of a conjecture involving unknown semialgebraic functions with specified smoothness. More precisely, we prove the following result: Let H be a semialgebraic bundle with respect to C m loc ( R 2 , R D ). If H has a section, then it has a semialgebraic section.

Let Ω be a domain in R d+1 , d≥1 . In the paper's references [HMM2] and [GMT] it was proved that if Ω satisfies a corkscrew condition and if ∂Ω is d -Ahlfors regular, i.e. Hausdorff measure H d (B(x,r)∩∂Ω)∼ r d for all x∈∂Ω and 0<r<diam(∂Ω) , then ∂Ω is uniformly rectifiable if and only if (a) a square function Carleson measure estimate holds for every bounded harmonic function on Ω or (b) an ε -approximation property for all 0<ε<1 for every such function. Here we explore (a) and (b) when ∂Ω is not required to be Ahlfors regular. We first prove that (a) and (b) hold for any domain Ω for which there exists a domain Ω ˜ ⊂Ω such that ∂Ω⊂∂ Ω ˜ and ∂ Ω ˜ is uniformly rectifiable. We next assume Ω satisfies a corkscrew condition and ∂Ω satisfies a capacity density condition. Under these assumptions we prove conversely that the existence of such Ω ˜ implies (a) and (b) hold on Ω and give further characterizations of domains for which (a) or (b) holds. One is that harmonic measure satisfies a Carleson packing condition for diameters similar to the corona decompositionm proved equivalent to uniform rectifiability in [GMT]. The second characterization is reminiscent of the Carleson measure description of H ∞ interpolating sequences in the unit disc.

Let P??Q p [x,y] , s?�C with sufficiently large real part, and consider the integral operator ( A P,s f)(y):= 1 1??p ?? ??Z p |P(x,y) | s f(x)|dx| on L 2 ( Z p ) . We show that if P is homogeneous then for each character ? of Z ? p the characteristic function det(1?�u A P,s,? ) of the restriction A P,s,? of A P,s to the eigenspace L 2 ( Z p ) ? is the q -Wronskian of a set of solutions of a (possibly confluent) q -hypergeometric equation. In particular, the nonzero eigenvalues of A P,s,? are the reciprocals of the zeros of such q -Wronskian.