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preprint2016arXiv

Endpoint Mapping properties of the Littlewood-Paley square function

In this note we give an alternative proof of a theorem due to Bourgain \cite{Bourgain} concerning the growth of the constant in the Littlewood-Paley inequality on $\mathbb{T}$ as $p \rightarrow 1^+$. Our argument is based on the endpoint mapping properties of Marcinkiewicz multiplier operators, obtained by Tao and Wright in \cite{TW}, and on Tao's converse extrapolation theorem \cite{Tao}. Our method also establishes the growth of the constant in the Littlewood-Paley inequality on $\mathbb{T}^n$ as $p \rightarrow 1^+$. Furthermore, we obtain sharp weak-type inequalities for the Littlewood-Paley square function on $\mathbb{T}^n$, but when $n \geq 2$ the weak-type endpoint estimate on the product Hardy space over the $n$-torus fails, contrary to what happens when $n=1$.

preprint2017arXiv

On weighted mixed-norm Sobolev estimates for some basic parabolic equations

Novel global weighted parabolic Sobolev estimates, weighted mixed-norm estimates and a.e. convergence results of singular integrals for evolution equations are obtained. Our results include the classical heat equation, the harmonic oscillator evolution equation $$\partial_tu=Δu-|x|^2u+f,$$ and their corresponding Cauchy problems. We also show weighted mixed-norm estimates for solutions to degenerate parabolic extension problems arising in connection with the fractional space-time nonlocal equations $(\partial_t-Δ)^su=f$ and $(\partial_t-Δ+|x|^2)^su=f$, for $0<s<1$.

preprint2016arXiv

Local Stable Manifold theorem for fractional systems revisited

The subject of fractional calculus has witnessed rapid development over past few decades. In particular the area of fractional differential equations has received considerable attention. Several theoretical results have been obtained and powerful numerical methods have been developed. In spite of the extensive numerical simulations that have been carried out in the area of fractional order dynamical systems, analytical results obtained are very few. In pursuance to this, present authors have extended local stable manifold theorem in case of fractional systems \cite{deshpande2016local}. Cong et al. \cite{cong2016stable} have pointed out discrepancies in the asymptotic expansion of two-parameter Mittag-Leffler functions with matrix argument (\textit{cf.} Lemma 4 part 2 of article \cite{deshpande2016local}). In the present communication we give the corrected expansion of the same and prove the local stable manifold theorem by following the same approach given in \cite{deshpande2016local}.

preprint2017arXiv

Restriction of the Fourier transform to some oscillating curves

Let $ϕ$ be a smooth function on a compact interval $I$. Let $$γ(t)=\left (t,t^2,\cdots,t^{n-1},ϕ(t)\right).$$ In this paper, we show that $$\left(\int_I \big|\hat f(γ(t))\big|^q \big|ϕ^{(n)}(t)\big|^{\frac{2}{n(n+1)}} dt\right)^{1/q}\le C\|f\|_{L^p(\mathbb R^n)}$$ holds in the range $$1\le p<\frac{n^2+n+2}{n^2+n},\quad 1\le q<\frac{2}{n^2+n}p'.$$ This generalizes an affine restriction theorem of Sjölin (1974) for $n=2$. Our proof relies on ideas of Sjölin (1974) and Drury (1985), and more recently Bak-Oberlin-Seeger (2008) and Stovall (2016), as well as a variation bound for smooth functions.

preprint2016arXiv

When the Cauchy inequality becomes a formula

In this note we revisit the classical geometric-arithmetic mean inequality and find a formula for the difference of the arithmetic and the geometric means of given $n\in\mathbb N$ nonnegative numbers $x_1,x_2,\dots,x_n$. The formula yields new stronger versions of the geometric-arithmetic mean inequality. We also find a second version of a strong geometric-arithmetic mean inequality and show that all inequalities are optimal in some sense. Anther striking novelty is, that the equality in all new inequalities holds not only in the case when all $n$ numbers are equal, but also in other cases.

preprint2016arXiv

A Note on Jain basis functions

In the present article the moments associated with the Jain basis functions are developed to order ten. The moments are shown to be polynomials in one variable with polynomial coefficients. The polynomial coefficients are investigated and shown to be linked to sequences found in the On-line Encyclopedia of Integer Sequences.

preprint2016arXiv

The Evans-Krylov theorem for nonlocal parabolic fully nonlinear equations

In this paper, we prove the Evans-Krylov theorem for nonlocal parabolic fully nonlinear equations.

preprint2017arXiv

Exact Hausdorff and Packing measure of certain Cantor sets, not necessarily self-similar or homogeneous

We compute the exact Hausdorff and Packing measures of linear Cantor sets which might not be self similar or homogeneous . The calculation is based on the local behavior of the natural probability measure supported on the sets.

preprint2016arXiv

Existence and uniqueness of the modified error function

This article is devoted to prove the existence and uniqueness of solution to the non-linear second order differential problem through which is defined the modified error function introduced in Cho-Sunderland, J. Heat Transfer, 96-2:214-217, 1974. We prove here that there exists a unique non-negative analytic solution for small positive values of the parameter on which the problem depends.

preprint2017arXiv

Joint eigenfunctions for the relativistic Calogero-Moser Hamiltonians of hyperbolic type. II. The two- and three-variable cases

In a previous paper we introduced and developed a recursive construction of joint eigenfunctions $J_N(a_+,a_-,b;x,y)$ for the Hamiltonians of the hyperbolic relativistic Calogero-Moser system with arbitrary particle number $N$. In this paper we focus on the cases $N=2$ and $N=3$, and establish a number of conjectured features of the corresponding joint eigenfunctions. More specifically, choosing $a_+,a_-$ positive, we prove that $J_2(b;x,y)$ and $J_3(b;x,y)$ extend to globally meromorphic functions that satisfy various invariance properties as well as a duality relation. We also obtain detailed information on the asymptotic behavior of similarity transformed functions E$_2(b;x,y)$ and E$_3(b;x,y)$. In particular, we determine the dominant asymptotics for $y_1-y_2\to\infty$ and $y_1-y_2,y_2-y_3\to\infty$, resp., from which the conjectured factorized scattering can be read off.

preprint2017arXiv

Differential Equations with a Difference Quotient

The purpose of this paper is to study a class of ill-posed differential equations. In some settings, these differential equations exhibit uniqueness but not existence, while in others they exhibit existence but not uniqueness. An example of such a differential equation is, for a polynomial $P$ and continuous functions $f(t,x):[0,1]\times [0,1]\rightarrow \mathbb{R}$, \begin{equation*} \frac{\partial}{\partial t} f(t,x) = \frac{ P(f(t,x))-P(f(t,0))}{x}, \quad x>0. \end{equation*} These differential equations are related to inverse problems.

preprint2016arXiv

On properties of the coefficients of the complicated and exotic formal solutions of the sixth Painlevé equation

It is known, that among the formal solutions of the sixth Painlevé equation there met series with integer power exponents of the independent variable $x$ with coefficients in form of formal Laurent series (with finite main parts) in $\log^{-1} x$ (complicated expansions), or in $x^{{\rm i}\,θ}$, where ${\rm i}=\sqrt{-1},$ $θ\in\mathbb{R},$ $θ\neq 0$ (exotic expansions). These coefficients can be computed consecutively. Here we research analytic properties of the series, that are the coefficients of the complicated and exotic formal solutions of the sixth Painlevé equation.

preprint2014arXiv

On $(α_n)$-regular sets

We define $(α_n)$ -regular sets in uniformly perfect metric spaces. This definition is quasisymmetrically invariant and the construction resembles generalized dyadic cubes in metric spaces. For these sets we then determine the necessary and sufficient conditions to be fat (or thin). In addition we discuss restrictions of doubling measures to these sets, and in particular give a sufficient condition to retain at least some of the restricted measures doubling on the set. Our main result generalizes and extends analogous results that were previously known to hold in the real-line.

preprint2016arXiv

Tangents, rectifiability, and corkscrew domains

In a recent paper, Csörnyei and Wilson prove that curves in Euclidean space of $σ$-finite length have tangents on a set of positive $\mathscr{H}^{1}$-measure. They also show that a higher dimensional analogue of this result is not possible without some additional assumptions. In this note, we show that if $Σ\subseteq \mathbb{R}^{d+1}$ has the property that each ball centered on $Σ$ contains two large balls in different components of $Σ^{c}$ and $Σ$ has $σ$-finite $\mathscr{H}^{d}$-measure, then it has $d$-dimensional tangent points in a set of positive $\mathscr{H}^{d}$-measure. We also give shorter proofs that Semmes surfaces are uniformly rectifiable and, if $Ω\subseteq \mathbb{R}^{d+1}$ is an exterior corkscrew domain whose boundary has locally finite $\mathscr{H}^{d}$-measure, one can find a Lipschitz subdomain intersecting a large portion of the boundary.

preprint2016arXiv

New fractional integral unifying six existing fractional integrals

In this paper we introduce a new fractional integral that generalizes six existing fractional integrals, namely, Riemann-Liouville, Hadamard, Erdélyi-Kober, Katugampola, Weyl and Liouville fractional integrals in to one form. Such a generalization takes the form \[ \left({}^ρ\mathcal{I}^{α, β}_{a+;η, κ}f\right)(x)=\frac{ρ^{1-β}x^κ}{Γ(α)}\int_a^x \frac{τ^{ρη+ρ-1}}{(x^ρ-τ^ρ)^{1-α}}f(τ)\text{d}τ, \quad 0\leq a < x < b \leq \infty. \] A similar generalization is not possible with the Erdélyi-Kober operator though there is a close resemblance with the operator in question. We also give semigroup, boundedness, shift and integration-by-parts formulas for completeness.

preprint2016arXiv

Doubling condition at the origin for non-negative positive definite functions

We study upper and lower estimates as well as the asymptotic behavior of the sharp constant $C=C_n(U,V)$ in the doubling-type condition at the origin \[ \frac{1}{|V|}\int_{V}f(x)\,dx\le C\,\frac{1}{|U|}\int_{U}f(x)\,dx, \] where $U,V\subset \mathbb{R}^{n}$ are $0$-symmetric convex bodies and $f$ is a non-negative positive definite function.

preprint2016arXiv

Convolution Operators with Singular Measures of Fractional Type on the Heisenberg Group

We study the type set of singular measures of fractional type on the Heisenbrg group.

preprint2016arXiv

Some integrals involving generalized k-Struve functions

The close form of some integrals involving recently developed generalized k-Struve functions is obtained. The outcome of these integrations is expressed in terms of generalized Wright functions. Several special cases are deduced which lead to some known results.

preprint2016arXiv

On fractional $q$-extensions of some $q$-orthogonal polynomials

Fractional $q$-extensions of some classical $q$-orthogonal polynomials are introduced and some of the main properties of the new defined functions are given. Next, a fractional $q$-difference equation of Gauss type is introduced and solved by means of power series method.

preprint2016arXiv

On computing differential transform of nonlinear non-autonomous functions and its applications

Although being powerful, the differential transform method yet suffers from a drawback which is how to compute the differential transform of nonlinear non-autonomous functions that can limit its applicability. In order to overcome this defect, we introduce in this paper a new general formula and its related recurrence relations for computing the differential transform of any analytic nonlinear non-autonomous function with one or multi-variable. Regarding, the formula in the literature was found not applicable to deal with the present non-autonomous functions. Accordingly, a generalization is presented in this paper which reduces to the corresponding formula in the literature as a special case. Several test examples for different types of nonlinear differential and integro-differential equations are solved to demonstrate the validity and applicability of the present method. The obtained results declare that the suggested approach not only effective but also a straight forward even in solving differential and integro-differential equations with complex nonlinearities.

preprint2016arXiv

Equality of the algebraic and geometric ranks of Cartan subalgebras and applications to linearization of a system of ordinary differential equations

If $L$ is a semisimple Lie algebra of vector fields on R^N with a split Cartan subalgebra C, then it is proved that the dimension of the generic orbit of C coincides with the dimension of C. As a consequence one obtains a local canonical form of L in terms of exponentials of coordinate functions and vector fields that are independent of these coordinates -- for a suitable choice of coordinates. This result is used to classify semisimple algebras of vector fields on R^3 and to determine all representations of sl(N, R) as vector fields on R^N. These representations are used to find linearizing coordinates for any second order ordinary differential equation that admits sl(3, R) as its symmetry algebra and for a system of two second order ordinary differential equations that admits sl(4, R) as its symmetry algebra.

preprint2016arXiv

Geodesic distance Riesz energy on the sphere

We study energy integrals and discrete energies on the sphere, in particular, analogs of the Riesz energy with the geodesic distance in place of Euclidean, and observe that the range of exponents for which the uniform distribution optimizes such energies is different from the classical case. We also obtain a general form of the Stolarsky principle, which relates discrete energies to certain $L^2$ discrepancies. This leads to new proofs of discrepancy estimates, as well as the sharp asymptotics of the difference between optimal discrete and continuous energies in the geodesic case.

preprint2016arXiv

On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight

We study the asymptotics of recurrence coefficients for monic orthogonal polynomials $π_n(z)$ with the quartic exponential weight $\exp[-N(\frac 12 z^2+\frac 14 tz^4)]$, where $t\in {\mathbb C}$ and $N\in{\mathbb N}$, $N\to\infty$. Our goal is to describe these asymptotic behaviors globally for $t\in {\mathbb C}$ in different regions. We also describe the "breaking" curves separating these regions, and discuss their special (critical) points. All these pieces of information combined provide the global asymptotic "phase portrait" of the recurrence coefficients of $π_n(z)$, which was studied numerically in [Constr. Approx. 41 (2015), 529-587, arXiv:1108.0321]. The main goal of the present paper is to provide a rigorous framework for the global asymptotic portrait through the nonlinear steepest descent analysis (with the $g$-function mechanism) of the corresponding Riemann-Hilbert problem (RHP) and the continuation in the parameter space principle. The latter allows to extend the nonlinear steepest descent analysis from some parts of the complex $t$-plane to all noncritical values of $t$. We also provide explicit solutions for recurrence coefficients in terms of the Ri

preprint2016arXiv

Elliptic hypergeometric functions

This is a brief overview of the status of the theory of elliptic hypergeometric functions to the end of 2012 written as a complementary chapter to the Russian edition of the book by G.E. Andrews, R. Askey, and R. Roy, Special Functions, Encycl. of Math. Appl. 71, Cambridge Univ. Press, 1999.

778 works