---
tag: 75WGJ014GR
kind: concept
name: Real Hardy space H1
msc: [42B30, 42B35]
sources:
  - url: https://en.wikipedia.org/wiki/Hardy_space
---

## Summary

The real Hardy space $H^1(\R^n)$ is the space of integrable functions $f$ on $\R^n$ whose Riesz transforms $R_1 f, \dots, R_n f$ are also integrable.
It is a proper subspace of $L^1(\R^n)$, and its elements have integral zero.
It is the right replacement for $L^1$ in harmonic analysis.
Singular integrals such as the Hilbert and Riesz transforms map $H^1$ to $L^1$, but not $L^1$ to $L^1$.
Its dual space is the space [[BMO|QJYTXFT8Y4]] of functions of bounded mean oscillation, and its elements are sums of simple building blocks called atoms.
For $n = 1$ it is the space of real parts of boundary values of functions in the holomorphic [[Hardy space|PNNNFJVF6W]] $H^1$ of the upper half-plane.

## Explanations

### L1 human

An integrable function can still behave badly under singular integrals, like the Hilbert transform, which can turn it into a non-integrable function.
The real Hardy space keeps exactly the integrable functions for which this does not happen.
Such functions must have some cancellation: their total integral is zero.
A typical element on $\R$ is $\mathbf{1}_{[0, 1]} - \mathbf{1}_{[-1, 0]}$, a positive bump next to a negative bump of the same size.