4.4 Riesz potentials

Riesz potentials generalize the Newtonian potential to higher dimensions and fractional orders. They're key in potential theory, PDEs, and harmonic analysis, defined as convolution integrals that smooth functions by averaging with a decaying kernel.

These potentials have important properties like boundedness, continuity, and decay at infinity. They're closely tied to function spaces like Sobolev and Bessel potential spaces, making them crucial for studying PDEs and regularity of solutions.

Definition of Riesz potentials

• Riesz potentials are fundamental objects in potential theory that generalize the classical Newtonian potential to higher dimensions and fractional orders
• Play a crucial role in the study of partial differential equations, harmonic analysis, and function spaces

Integral representation

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• Riesz potential of order $\alpha$ of a function $f$ in $\mathbb{R}^n$ is defined as the convolution integral: $I_\alpha f(x) = \frac{1}{\gamma(\alpha)} \int_{\mathbb{R}^n} \frac{f(y)}{|x-y|^{n-\alpha}} dy$ where $\gamma(\alpha)$ is a normalization constant and $0 < \alpha < n$
• Singular integral operator that averages the values of $f$ with a kernel that decays like $|x-y|^{-(n-\alpha)}$
• Represents a smoothing operation that increases the regularity of the function $f$

Relation to Newtonian potential

• In the case $n=3$ and $\alpha=2$, the Riesz potential reduces to the Newtonian potential (up to a constant factor): $I_2 f(x) = \frac{1}{4\pi} \int_{\mathbb{R}^3} \frac{f(y)}{|x-y|} dy$
• Newtonian potential is the fundamental solution of the Laplace equation in $\mathbb{R}^3$ and plays a key role in classical potential theory and electrostatics
• Riesz potentials provide a natural generalization of the Newtonian potential to higher dimensions and fractional orders

Generalization to higher dimensions

• Riesz potentials can be defined in any dimension $n \geq 1$ and for any order $\alpha$ satisfying $0 < \alpha < n$
• Enable the study of potential-theoretic problems and PDEs in higher-dimensional spaces ($\mathbb{R}^n$, $n > 3$)
• Higher-dimensional Riesz potentials have applications in mathematical physics, such as in the study of higher-dimensional electrostatics and quantum mechanics

Properties of Riesz potentials

• Riesz potentials possess several important properties that make them useful tools in potential theory and related fields
• Properties depend on the order $\alpha$ and the dimension $n$ of the underlying space

Boundedness and integrability

• Riesz potentials are bounded operators from $L^p(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n)$ for suitable exponents $p$ and $q$ satisfying $1 < p < q < \infty$ and $\frac{1}{q} = \frac{1}{p} - \frac{\alpha}{n}$
• Integrability of Riesz potentials depends on the integrability of the function $f$ and the order $\alpha$
  • If $f \in L^p(\mathbb{R}^n)$ with $1 \leq p < \frac{n}{\alpha}$, then $I_\alpha f \in L^q(\mathbb{R}^n)$ with $\frac{1}{q} = \frac{1}{p} - \frac{\alpha}{n}$
  • If $f \in L^1(\mathbb{R}^n)$, then $I_\alpha f$ is locally integrable for any $0 < \alpha < n$

Continuity and differentiability

• Riesz potentials are continuous operators from $L^p(\mathbb{R}^n)$ to $C^{k,\gamma}(\mathbb{R}^n)$ (space of $k$-times continuously differentiable functions with Hölder continuous $k$-th derivatives of exponent $\gamma$) for suitable $p$, $k$, and $\gamma$
• Differentiability of Riesz potentials depends on the differentiability of the function $f$ and the order $\alpha$
  • If $f \in C^k(\mathbb{R}^n)$, then $I_\alpha f \in C^{k+[\alpha]}(\mathbb{R}^n)$, where $[\alpha]$ denotes the integer part of $\alpha$
  • If $f \in C^{\infty}(\mathbb{R}^n)$, then $I_\alpha f \in C^{\infty}(\mathbb{R}^n)$ for any $0 < \alpha < n$

Behavior at infinity

• Riesz potentials exhibit a certain decay behavior at infinity, depending on the integrability of the function $f$ and the order $\alpha$
  • If $f \in L^p(\mathbb{R}^n)$ with $1 \leq p < \frac{n}{\alpha}$, then $I_\alpha f(x) = O(|x|^{\alpha-n})$ as $|x| \to \infty$
  • If $f \in L^1(\mathbb{R}^n)$, then $I_\alpha f(x) = o(1)$ as $|x| \to \infty$ for any $0 < \alpha < n$
• Decay properties of Riesz potentials are crucial in the study of the asymptotic behavior of solutions to PDEs and in potential theory

Semigroup property

• Riesz potentials satisfy the semigroup property: for any $\alpha, \beta > 0$ with $\alpha + \beta < n$, we have $I_\alpha (I_\beta f) = I_{\alpha+\beta} f$
• Semigroup property allows the composition of Riesz potentials of different orders
• Useful in the study of fractional differential equations and in the theory of function spaces

Riesz potentials and function spaces

• Riesz potentials are closely related to various function spaces in harmonic analysis and PDEs
• Mapping properties of Riesz potentials between different function spaces provide a powerful tool for studying the regularity of solutions to PDEs

Lp spaces and Riesz potentials

• Riesz potentials are bounded operators from $L^p(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n)$ for suitable exponents $p$ and $q$ satisfying $1 < p < q < \infty$ and $\frac{1}{q} = \frac{1}{p} - \frac{\alpha}{n}$
• Mapping properties of Riesz potentials in $L^p$ spaces are characterized by the Hardy-Littlewood-Sobolev inequality
• $L^p$-$L^q$ estimates for Riesz potentials are fundamental in the study of elliptic and parabolic PDEs

Sobolev spaces and Riesz potentials

• Riesz potentials are closely related to Sobolev spaces $W^{k,p}(\mathbb{R}^n)$ (space of functions with weak derivatives up to order $k$ in $L^p(\mathbb{R}^n)$)
• For $k \in \mathbb{N}$ and $1 < p < \infty$, the Riesz potential $I_k$ is an isomorphism between $L^p(\mathbb{R}^n)$ and $W^{k,p}(\mathbb{R}^n)$
• Sobolev embedding theorems can be proved using the mapping properties of Riesz potentials

Bessel potential spaces

• Bessel potential spaces $H^{s,p}(\mathbb{R}^n)$ are a generalization of Sobolev spaces that allow for fractional orders of differentiability $s \in \mathbb{R}$
• Bessel potentials $G_s = (I - \Delta)^{-s/2}$ are defined using the Fourier transform and are related to Riesz potentials via the identity $G_s = I_s * G_0$
• Mapping properties of Bessel potentials and their relation to Riesz potentials are essential in the theory of function spaces and in the study of PDEs

Triebel-Lizorkin spaces

• Triebel-Lizorkin spaces $F^{s}_{p,q}(\mathbb{R}^n)$ are a family of function spaces that generalize Sobolev and Bessel potential spaces and provide a finer scale of regularity
• Riesz potentials and their mapping properties play a crucial role in the theory of Triebel-Lizorkin spaces
• Characterizations of Triebel-Lizorkin spaces using Riesz potentials and their connections to other function spaces are important in harmonic analysis and PDEs

Riesz potentials and PDEs

• Riesz potentials are fundamental tools in the study of partial differential equations (PDEs)
• Mapping properties of Riesz potentials and their relation to various PDEs provide insights into the existence, uniqueness, and regularity of solutions

Laplace equation and Riesz potentials

• Riesz potentials are closely related to the Laplace equation $\Delta u = 0$ in $\mathbb{R}^n$
• In the case $n=3$ and $\alpha=2$, the Riesz potential $I_2$ is the fundamental solution of the Laplace equation (up to a constant factor)
• Properties of Riesz potentials can be used to study the behavior of harmonic functions (solutions to the Laplace equation) and their derivatives

Poisson equation and Riesz potentials

• Riesz potentials are also connected to the Poisson equation $\Delta u = f$ in $\mathbb{R}^n$
• Solution to the Poisson equation can be expressed using the Riesz potential of the right-hand side $f$: $u = I_2 f$
• Mapping properties of Riesz potentials provide regularity results for solutions to the Poisson equation

Fractional Laplacian and Riesz potentials

• Fractional Laplacian $(-\Delta)^{s/2}$ is a non-local operator that generalizes the classical Laplacian to fractional orders $s \in (0,2)$
• Riesz potentials are closely related to the inverse of the fractional Laplacian: $(-\Delta)^{-s/2} = I_s$
• Properties of Riesz potentials are crucial in the study of fractional differential equations involving the fractional Laplacian

Riesz potentials in harmonic analysis

• Riesz potentials play a central role in harmonic analysis, particularly in the study of singular integrals and their applications to PDEs
• Calderon-Zygmund theory of singular integrals heavily relies on the properties of Riesz potentials
• Riesz transforms, which are singular integral operators related to Riesz potentials, are fundamental in the study of harmonic functions and in the theory of Hardy spaces

Applications of Riesz potentials

• Riesz potentials find numerous applications in various branches of mathematics and mathematical physics
• Mapping properties and the relation of Riesz potentials to PDEs make them valuable tools in modeling and analyzing physical phenomena

Potential theory and Riesz potentials

• Riesz potentials are fundamental objects in potential theory, which studies the behavior of harmonic functions and their generalizations
• Properties of Riesz potentials are used to investigate the boundary behavior of potentials, capacity, and thin sets
• Riesz potentials are also employed in the study of the Dirichlet problem and the Newtonian potential in higher dimensions

Riesz potentials in electrostatics

• In electrostatics, Riesz potentials are used to model the electric potential generated by a charge distribution
• Newtonian potential, a special case of Riesz potentials, represents the electric potential due to a volume charge density in three dimensions
• Mapping properties of Riesz potentials are utilized to study the regularity of the electric field and the behavior of the potential at infinity

Riesz potentials in fluid dynamics

• Riesz potentials appear in the study of fluid dynamics, particularly in the context of the Navier-Stokes equations
• Biot-Savart law, which relates the velocity field of an incompressible fluid to its vorticity, can be expressed using Riesz potentials
• Properties of Riesz potentials are employed to investigate the regularity of solutions to the Navier-Stokes equations and to study the behavior of vorticity

Riesz potentials in quantum mechanics

• In quantum mechanics, Riesz potentials are used to model the interaction between particles
• Yukawa potential, which describes the short-range interaction between nucleons, is a special case of Riesz potentials
• Properties of Riesz potentials are utilized to study the scattering theory and the bound states of quantum systems

Advanced topics in Riesz potentials

• Several advanced topics in the theory of Riesz potentials have been the subject of active research in recent years
• These topics involve the generalization of Riesz potentials to various settings and their connections to other areas of mathematics

Riesz potentials on manifolds

• Riesz potentials can be generalized to Riemannian manifolds, where they are defined using the heat kernel or the distance function of the manifold
• Properties of Riesz potentials on manifolds, such as their mapping properties and their relation to the Laplace-Beltrami operator, are studied in geometric analysis
• Riesz potentials on manifolds find applications in the study of harmonic functions, heat equations, and spectral theory on manifolds

Riesz potentials and fractional calculus

• Riesz potentials are closely related to fractional calculus, which studies derivatives and integrals of arbitrary order
• Fractional Laplacian, a fundamental operator in fractional calculus, can be expressed using Riesz potentials
• Properties of Riesz potentials are employed to study the mapping properties of fractional differential operators and to investigate the regularity of solutions to fractional differential equations

Riesz potentials and singular integrals

• Riesz potentials are deeply connected to the theory of singular integrals, which are operators that involve principal value integrals and have a singularity in their kernel
• Riesz transforms, which are singular integral operators related to Riesz potentials, play a crucial role in harmonic analysis and in the study of PDEs
• Mapping properties of Riesz potentials and their relation to singular integrals are essential in the development of Calderon-Zygmund theory and its applications

Nonlinear Riesz potentials

• Nonlinear generalizations of Riesz potentials, such as the p-Laplace equation and the fractional p-Laplacian, have been the subject of extensive research in recent years
• These nonlinear potentials are related to quasilinear PDEs and have applications in various fields, such as image processing and phase transitions
• Properties of nonlinear Riesz potentials, such as their mapping properties and their relation to Sobolev spaces, are studied using techniques from nonlinear analysis and calculus of variations