In regard to $L^p$ spaces, is $L^p \subset L^q$ for $p

Question

Attempting to understand if there are relationships between $L^p$ spaces and can't find a conclusive answer in Kreyszig's 'Introduction to Functional Analysis with Applications' nor in a solid browsing of Wikipedia.

More context:

Dirichlet conditions seem to indicate that if a function is absolutely integrable it can be represented as a fourier series. To me, this means that the function must lie within $L^1$ space, i.e. $\int | f(x) | \ dx < \infty$.

I began this hunt in an attempt to understand why functions in $L^2([0,1])$ space can be represented as fourier series, or more specifically, why the basis components of $L^2([0,1])$ are functions of $x$ (e.g. $e^{-2 \pi i n x}$). If $L^2([0,1])$ functions are contained within $L^1$ then I think I understand, though this seems like a leap.

Appreciate any help.

Answer 1

As @Thorgott commented, there is no relation between those spaces in general, but $$ L^p(\Omega)\subset L^q(\Omega)\qquad\textrm{for}\quad p>q>0, $$ if $\Omega$ has finite measure, since the Hölder inequality gives $$ \|u\|_{L^q(\Omega)}^q =\||u|^q\|_{L^1(\Omega)} \leq \|1\|_{L^{p/(p-q)}(\Omega)} \||u|^q\|_{L^{p/q}(\Omega)} =|\Omega|^{(p-q)/p} \|u\|_{L^{p}(\Omega)}^q. $$ In fact, we do not need the restriction $p,q\geq1$.

To illustrate what happens when $|\Omega|=\infty$, take $\Omega=\mathbb{R}$ with the Lebesgue measure on it. Then the function $u\equiv1$ is in $L^\infty(\mathbb{R})\setminus L^1(\mathbb{R})$. However, the function $u(x)=\min\{0,\log|x|$} is in $L^1(\mathbb{R})\setminus L^\infty(\mathbb{R})$.

Answer 2

Alex, when reading the title it looks like you ask for conditions to inclusion between $L^p$-spaces, but in your query, it looks more like you are into the convergence of Fourier series. Since these matters are different I split the answer into two parts.

Inclusion between $L^p$-spaces

When the measure space, i.e. the domain of integration, is

1. Finite (like $[0,1]$), then as @timur pointed out, we have $$L^p\subset L^q, \quad \text{provided $p>q>0$} $$

2. Infinite and non-discrete, then as @timur pointed out there is no inclusion.

3. Infinite and discrete, then $$L^p\subset L^q, \quad \text{provided $q>p>0$} $$
   see How do you show monotonicity of the $\ell^p$ norms?

These matters are connected to the Fourier series in a theorem known as the Pontryagin duality. The interval $[0,1]$ can be viewed upon as the circle group, and the dual group of the circle group is the group of integers (a discrete infinite group) - in case of the real numbers, the dual group is the real numbers again.

Convergence of Fourier series

Fourier series are series of the form $f(x)\sim\sum_{n=-\infty}^\infty c_ne^{2\pi inx}$, with $c_n=\int_0^1f(x)e^{-2\pi inx}dx$. Such series are defined for $f\in L^p[0,1]$ when $p\geq1$. When talking about convergence of Fourier series one usually means pointwise convergence (almost everywhere since sets of zero measures are unimportant in the Lebesgue sense).

Kolmogorov constructed a function in $L^1$ where the Fourier series, which do exist, diverges almost everywhere (1922).

Note that Kolmogorov's example shows that extra conditions (like the Dini-test) upon $L^1$-functions to have convergent Fourier series are non-superfluous. For long it was an open problem if the Fourier series of a $L^2$-function converge or not, and it was proved to be true by Carleson in 1962, and was later sharpened to $L^p$ $1<p\leq2$ by Hunt.

Interestingly, it could be mentioned that there are intermediate spaces $A$ and $B$, with $L^1\supset A\supset B\supset L^p$ for all $p>1$, where $A$ contain functions with divergent Fourier series and while $B$ does not which has been studied by Konyagin and others.