Weighted norm inequalities for convolutions, differential operators, and generalized hypergeometric functions
A sharp weighted norm inequalities for the integral convolutions \([f*g](a,x)=\int_a^x f(t)g(x-t) dt\), \(x\in [a,b]\), and applications are discussed in this paper. The following theorem gives an inequality for the convolution on a finite interval and it can be used as a generator of various integro-differential and special function inequalities. Theorem 3. Let \(f\) and \(g\) be measurable complex-valued functions which are defined for a.e. \(x\) in \([a,b]\) and \([0,b-a]\), respectively (\(-\infty <a<b<\infty\)). Then the following inequality holds for any numbers \(\alpha, \beta, \lambda >0\), \(p>1\) (\(1/p+1/q=1\)), and \(\tau \in (0,\min(p,q))\), provided that the integrals are finite: \[ \left[\int_a^b (x-a)^{(\alpha+\beta-1)(1-\tau)} (b-x)^{\lambda-1}|[f*g](a,x)|^\tau dx \right]^{1/\tau} \] \[ \leq K (b-a)^{(\alpha+\beta+\lambda-1)(1/\tau -1)} \left[\int_a^b (x-a)^{(\alpha-1)(1-p)} (b-x)^{\beta+\lambda-1} |f(x)|^p dx \right]^{1/p} \] \[ \times \left[\int_0^{b-a} t^{(\beta-1)(1-q)} (b-a-t)^{\alpha+\lambda-1} |g(t)|^q dt \right]^{1/q}, \eqno(1) \] where \(K=K(\alpha, \beta, \lambda,p,\tau)\) is defined as \[ K=[\Gamma(\lambda)]^{1/\tau}\cdot \left[ \frac{\Gamma(\alpha)}{\Gamma(\alpha+\lambda)}\right]^{1/q} \cdot \left[ \frac{\Gamma(\beta)}{\Gamma(\beta+\lambda)}\right]^{1/p} \cdot \left[ \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha+\beta+\lambda)}\right]^{1/\tau -1}. \] If \(f\) and \(g\) are not identically \(0\) and \(x^{1-\alpha}f(a+(b-a)x)\) and \(x^{1-\beta} g((b-a)x)\) are continuous functions on \([0,1]\), then equality holds if and only if \(f(x)=c_1 (x-a)^{\alpha-1} e^{i\theta x}\) \((a\leq x\leq b)\) and \(g(x)=c_2 x^{\beta-1} e^{i\theta x}\) \((0\leq x\leq b-a)\), where \(\theta\) is any real number and \(c_1,c_2\) are any non-zero complex numbers. Theorem 3 allows to obtain various solution-kernel estimates for convolution integral equations. Namely, if functions \(g\) and \(h\) are given and the right-hand side of the first kind convolution Volterra equation is \[ [f*g](a,x) =h(x) \] then the inequality (1) can be considered as an integral estimate for any solution \(f\). The following theorem gives a general weighted norm inequality for the linear differential operators. Theorem 4. For \(n\geq 1\) let \(h(x)\) be a non-constant \(n\)-times continuously differentiable function on \([a,b]\) (\(-\infty <a<b<\infty\)) such that: \(h(a)=h'(a)=\ldots =h^{(n-1)}(a)=0\). Also let \(D[h(x)]=\sum_{k=0}^{n-1} a_kh^{(k)}(x)\) and \(M[h(x)]=\sum_{k=0}^n c_k h^{(k)}(x)\), where the coefficients \(a_k\) and \(c_k\) are any complex numbers (at least one coefficient \(a_k \not= 0\) and \(c_k\not= 0\)). Then the following inequality holds for any numbers \(\alpha, \beta, \lambda >0\), \(p>1\) (\(1/p+1/q=1\)), and \(\tau \in (0,\min(p,q))\), provided that the integrals are finite: \[ \left[\int_a^b (x-a)^{(\alpha+\beta-1)(1-\tau)} (b-x)^{\lambda-1}|D[h(x)]|^\tau dx \right]^{1/\tau} \] \[ \leq K_{DM} \left[\int_a^b (x-a)^{(\alpha-1)(1-p)} (b-x)^{\beta+\lambda-1} |M[h(x)]|^p dx \right]^{1/p}, \] where \[ K_{DM} = K (b-a)^{(\alpha+\beta+\lambda-1)(1/\tau -1)} \left[\int_0^{b-a} t^{(\beta-1)(1-q)} (b-a-t)^{\alpha+\lambda-1} |g(t)|^q dt \right]^{1/q}, \] \(K\) as defined in Theorem 3, and \(g\) is the solution to the linear differential equation \[ \sum_{k=0}^n c_k g^{(k)}(x)=0, \;\;x\in [0,b-a], \] which satisfies the initial conditions: \(g(0)=a_{n-1}/c_n\) and \(g^{(k)}(0)=[a_{n-1-k}-\sum_{l=0}^{k-1} c_{n-k+l} g^{(l)}(0)]/c_n\) for \(k=1,\ldots ,n-1\). The case of equality is also describe in a theorem. In the same section some examples involving the relatively simple kernels \(g\) and the corresponding solutions of Heaviside type are presented. The second part of the paper (Sections 5 and 6) consists of some weighted \(L^p\) norm inequalities involving generalized hypergeometric functions which are consequences of results from the article [\textit{A. Z. Grinshpan}, Adv. Appl. Math., 45, No.4, 564--606 (2010; Zbl 1220.26015)]. Let us mention one of those results. But before that, we have to describe the notations used in it. The generalized hypergeometric function \(_jF_k\) is defined by \[ _jF_k (\alpha_1, \ldots ,\alpha_j;\beta_1,\ldots ,\beta_k;z)=\sum_{n=0}^{\infty} \frac{\prod_{1\leq l\leq j}(\alpha_l)_n}{\prod_{1\leq l\leq k}(\beta_l)_n} \cdot \frac{z^n}{n!}, \;(j,k=0,1,\ldots ), \] where \((\alpha)_n=\alpha(\alpha+1)\cdots (\alpha +n-1)\) for \(n\geq 1\) and \((\alpha)_0=1\). If \(F(z)=\sum_{n=0}^\infty a_n z^n\) and \(\alpha >0\), then \(F_{*\alpha}\) is defined as \[ F_{*\alpha}(z)=\sum_{n=0}^\infty \frac{a_n}{(\alpha)_n}z^n. \;\;\;\quad\eqno(2) \] A normalized weighted \(L^p\) norm is defined as \[ || f||_{[p;\alpha,\beta,\gamma]} =\left[\int_0^1 t^{\alpha -1}(1-t)^{\beta -1} e^{\gamma t}|f(t)|^p \frac{dt}{B(\alpha,\beta)} \right]^{1/p}. \] Corollary 10. Let \(f(z)=1+a_1z+\cdots \) be analytic in the neighbourhood of the origin. Then for any \(\alpha, \beta, \lambda <0\), \(p>1\) (\(1/p+1/q=1\)), \(\tau \in (0,\min(p,q)]\), a complex \(\zeta\) and a real \(x\), the following inequality holds \[ [_1F_1(\alpha+\beta,\alpha +\beta+\lambda; \tau x)]^{1/\tau} \leq || f_{*\alpha}(\zeta t)||_{[p;\alpha,\beta+\lambda,px]} \cdot || (1/f)_{*\beta}(\zeta t)||_{[q;\beta,\alpha+\lambda,qx]}, \] where convolutions \(f_{*\alpha}\) and \((1/f)_{*\beta}\) are defined as in (2). For \(\zeta \not= 0\), the equality holds if and only if \(f\) is identically 1 in a neighbourhood of the origin and \(x=0\).
weighted inequalities for sums and integrals
scientific article; zbMATH DE number 6153927
scientific article