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Young's Inequality For Integral Operators
In mathematical analysis, the Young's inequality for integral operators, is a bound on the L^p\to L^q operator norm of an integral operator in terms of L^r norms of the kernel itself. Statement Assume that X and Y are measurable spaces, K : X \times Y \to \mathbb is measurable and q,p,r\geq 1 are such that \frac = \frac + \frac -1. If : \int_ , K (x, y), ^r \,\mathrm y \le C^r for all x\in X and : \int_ , K (x, y), ^r \,\mathrm x \le C^r for all y\in Y then Theorem 0.3.1 in: C. D. Sogge, ''Fourier integral in classical analysis'', Cambridge University Press, 1993. : \int_ \left, \int_ K (x, y) f(y) \,\mathrm y\^q \, \mathrm x \le C^q \left( \int_ , f(y), ^p \,\mathrm y\right)^\frac. Particular cases Convolution kernel If X = Y = \mathbb^d and K (x, y) = h (x - y) , then the inequality becomes Young's convolution inequality. See also Young's inequality for products In mathematics, Young's inequality for products is a mathematical inequality about the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Introduction and definition Given two normed vector spaces V and W (over the same base field, either the real numbers \R or the complex numbers \Complex), a linear map A : V \to W is continuous if and only if there exists a real number c such that \, Av\, \leq c \, v\, \quad \mbox v\in V. The norm on the left is the one in W and the norm on the right is the one in V. Intuitively, the continuous operator A never increases the length of any vector by more than a factor of c. Thus the image of a bounded set under a continuous operator is also bounded. Because of this property, the continuous linear operators are also known as bounded operators. In order to "measure the size" of A, one can take the infimum of the numbers c such that the abo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Operator
An integral operator is an operator that involves integration. Special instances are: * The operator of integration itself, denoted by the integral symbol * Integral linear operators, which are linear operators induced by bilinear forms involving integrals * Integral transforms, which are maps between two function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...s, which involve integrals {{mathanalysis-stub Integral calculus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christopher D
Christopher is the English version of a Europe-wide name derived from the Greek name Χριστόφορος (''Christophoros'' or '' Christoforos''). The constituent parts are Χριστός (''Christós''), "Christ" or " Anointed", and φέρειν (''phérein''), "to bear"; hence the "Christ-bearer". As a given name, 'Christopher' has been in use since the 10th century. In English, Christopher may be abbreviated as " Chris", "Topher", and sometimes "Kit". It was frequently the most popular male first name in the United Kingdom, having been in the top twenty in England and Wales from the 1940s until 1995, although it has since dropped out of the top 100. The name is most common in England and not so common in Wales, Scotland, or Ireland. People with the given name Antiquity and Middle Ages * Saint Christopher (died 251), saint venerated by Catholics and Orthodox Christians * Christopher (Domestic of the Schools) (fl. 870s), Byzantine general * Christopher Lekapenos (died 931) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Young's Convolution Inequality
In mathematics, Young's convolution inequality is a mathematical inequality about the convolution of two functions, named after William Henry Young. Statement Euclidean Space In real analysis, the following result is called Young's convolution inequality: Suppose f is in the Lebesgue spaceL^p(\Reals^d) and g is in L^q(\Reals^d) and \frac + \frac = \frac + 1 with 1 \leq p, q, r \leq \infty. Then \, f * g\, _r \leq \, f\, _p \, g\, _q. Here the star denotes convolution, L^p is Lebesgue space, and \, f\, _p = \Bigl(\int_ , f(x), ^p\,dx \Bigr)^ denotes the usual L^p norm. Equivalently, if p, q, r \geq 1 and \frac + \frac + \frac = 2 then \left, \int_ \int_ f(x) g(x - y) h(y) \,\mathrmx \,\mathrmy \ \leq \left(\int_ \vert f\vert^p\right)^\frac \left(\int_ \vert g\vert^q\right)^\frac \left(\int_ \vert h\vert^r\right)^\frac Generalizations Young's convolution inequality has a natural generalization in which we replace \Reals^d by a unimodular group G. If we let \mu be a bi-inv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Young's Inequality For Products
In mathematics, Young's inequality for products is a mathematical inequality about the product of two numbers. The inequality is named after William Henry Young and should not be confused with Young's convolution inequality. Young's inequality for products can be used to prove Hölder's inequality. It is also widely used to estimate the norm of nonlinear terms in PDE theory, since it allows one to estimate a product of two terms by a sum of the same terms raised to a power and scaled. Standard version for conjugate Hölder exponents The standard form of the inequality is the following: It can be used to prove Hölder's inequality. This form of Young's inequality can also be proved via Jensen's inequality. Young's inequality may equivalently be written as a^\alpha b^\beta \leq \alpha a + \beta b, \qquad\, 0 \leq \alpha, \beta \leq 1, \quad\ \alpha + \beta = 1. Where this is just the concavity of the logarithm function. Equality holds if and only if a = b or \ = \. G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
