Besov Spaces — Mathematics Knowledge System

Overview

Besov spaces B^s_{p,q}(R^n) are a scale of function spaces that interpolate between Sobolev spaces and provide precise control of smoothness. The parameter s is the smoothness exponent, p controls integrability, and q is a fine-tuning parameter. Besov spaces arise naturally as real interpolation spaces between Sobolev spaces: B^s_{p,q} = (W^{k,p}, W^{m,p})_{theta,q} for appropriate theta. They contain the Sobolev spaces W^{s,p} and Holder spaces C^s as special cases, and are essential for sharp embeddings, trace theorems, and the theory of elliptic PDE.

Intuition

Sobolev spaces W^{k,p} measure smoothness by requiring derivatives of order up to k to be in L^p. Besov spaces allow fractional smoothness s and add a second exponent q that distinguishes between functions with the same 'average' smoothness but different fine structure. The definition uses differences: a function has Besov smoothness s if its s-th order differences are in L^p in a quantitative way measured by the q-norm. This is like asking that the function is smooth in a 'Littlewood–Paley block' decomposition.

Formal Definition

Definition

The first-order modulus of continuity is omega_1(f,t)_p = sup_{|h|<=t} ||Delta_h f||_p where Delta_h f(x) = f(x+h) - f(x). For integer m > s, the Besov space B^s_{p,q}(R^n) consists of functions f in L^p with finite Besov norm. Using the real interpolation method: B^s_{p,q} = (W^{k_0,p}, W^{k_1,p})_{theta,q} where s = (1-theta)k_0 + theta k_1. Equivalently, via Littlewood–Paley decomposition: if f = sum_j phi_j * f (dyadic blocks), then ||f||_{B^s_{p,q}} = (sum_j 2^{jsq} ||phi_j * f||_p^q)^{1/q}.

\|f\|_{B^s_{p,q}} = \left(\|f\|_{L^p}^q + \int_0^1 \left(t^{-s}\omega_m(f,t)_p\right)^q \frac{dt}{t}\right)^{1/q}

Besov norm (modulus of continuity definition)

B^s_{p,q} = (W^{k_0,p}, W^{k_1,p})_{\theta,q},\quad s = (1-\theta)k_0 + \theta k_1

Interpolation characterisation

\|f\|_{B^s_{p,q}} \approx \left(\sum_{j=0}^\infty 2^{jsq}\|\phi_j * f\|_{L^p}^q\right)^{1/q}

Littlewood–Paley characterisation

Notation

Notation	Meaning
$B^s_{p,q}(\mathbb{R}^n)$	Besov space with smoothness s, integrability p, fine-tuning q
$\omega_m(f,t)_p$	m-th order modulus of continuity in L^p
$W^{s,p}$	Sobolev space (B^s_{p,2} for integer s)
$C^s$	Holder space (B^s_{inf,inf})

Theorems

Theorem 15.1: Theorem 15.1 (Special cases)

B^s_{p,2} = W^{s,p} for s a positive integer (Sobolev spaces). B^s_{inf,inf} = C^s (Holder spaces for non-integer s and Lipschitz spaces for integer s). B^{n/p}_{p,1} embeds continuously into L^inf (Besov embedding). B^0_{2,2} = L^2.

Theorem 15.2: Theorem 15.2 (Sobolev-type embedding)

For s > t and 1 <= p <= r <= infinity with s - n/p = t - n/r (same Sobolev exponent), B^s_{p,q}(R^n) embeds continuously into B^t_{r,q}(R^n). In particular B^s_{p,q} embeds into L^r for s = n/p - n/r > 0.

Theorem 15.3: Theorem 15.3 (Trace theorem)

The trace operator (restriction to a hyperplane) maps B^s_{p,q}(R^n) boundedly onto B^{s-1/p}_{p,q}(R^{n-1}) for s > 1/p. This is sharper than the Sobolev trace theorem, which loses log-regularity in the borderline case.

Worked Examples

1
Using the Littlewood–Paley characterisation: ||f||_{B^s_{2,2}}^2 = sum_j 2^{2js} ||phi_j*f||_{L^2}^2.
2
By Parseval, ||phi_j*f||_{L^2}^2 = ||hat(phi_j) hat(f)||_{L^2}^2 ~ integral_{2^j <= |xi| < 2^{j+1}} |hat(f)(xi)|^2 d xi.
3
Summing: ||f||_{B^s_{2,2}}^2 ~ sum_j 2^{2js} integral_{2^j <= |xi| < 2^{j+1}} |hat(f)|^2 d xi ~ integral (1+|xi|^2)^s |hat(f)(xi)|^2 d xi.
$\|f\|_{B^s_{2,2}}^2 \approx \int_{\mathbb{R}^n}(1+|\xi|^2)^s|\hat{f}(\xi)|^2\,d\xi = \|f\|_{H^s}^2$
4
So B^s_{2,2} = H^s with equivalent norms.

✓ Answer

B^s_{2,2}(R^n) = H^s(R^n) (Sobolev space) for all s >= 0, with equivalent norms via the Fourier characterisation.

Practice Problems

Hardfree response

Explain why B^s_{p,1} embeds into L^inf for s = n/p (the borderline Sobolev case) while W^{n/p,p} does not.

HardMultiple choice

Which inclusion is CORRECT for Besov spaces with fixed s and p?

B^s_{p,1} subset B^s_{p,2} subset B^s_{p,inf}B^s_{p,inf} subset B^s_{p,2} subset B^s_{p,1}B^s_{p,2} subset B^s_{p,1}All B^s_{p,q} are equal for different q

Quiz

Summary

Besov spaces B^s_{p,q} refine Sobolev spaces with a three-parameter scale (s, p, q).
Special cases: B^s_{2,2} = H^s (Sobolev), B^s_{inf,inf} = C^s (Holder), B^0_{2,2} = L^2.
Real interpolation: B^s_{p,q} = (W^{k,p}, W^{m,p})_{theta,q} with s = (1-theta)k + theta m.
Littlewood–Paley: ||f||_{B^s_{p,q}} ~ (sum_j 2^{jsq}||phi_j*f||_p^q)^{1/q}.
Inclusion: B^s_{p,1} subset B^s_{p,2} subset B^s_{p,inf}; larger q = larger (less restrictive) space.

References

BookTriebel, H. — Theory of Function Spaces, Birkhauser, 1983
BookBergh, J. & Lofstrom, J. — Interpolation Spaces: An Introduction, Springer, 1976
Websiteen.wikipedia.org