<p>In this paper, we consider analytic Hardy spaces associated with fractal domains in the complex plane. In 2013, Dong et al. obtained a positive solution to the Cantor set conjecture in (Adv Math 232:543–570, 2013) for the Cauchy transforms on the Sierpinski triangle. In this paper, we achieve a deeper understanding of these Cauchy transforms by showing that they do not belong to any Hardy space on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\triangle ^*,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>▵</mi> <mo>∗</mo> </msup> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\triangle ^*= \widehat{{\mathbb {C}}}\setminus \triangle \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>▵</mi> <mo>∗</mo> </msup> <mo>=</mo> <mover accent="true"> <mi mathvariant="double-struck">C</mi> <mo stretchy="true">^</mo> </mover> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mi>▵</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\triangle \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>▵</mi> </math></EquationSource> </InlineEquation> is the compact regular triangle with vertexes <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\{\varepsilon _k=e^{2k\pi i/3},\ k=0,1,2\}.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>ε</mi> <mi>k</mi> </msub> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mn>2</mn> <mi>k</mi> <mi>π</mi> <mi>i</mi> <mo stretchy="false">/</mo> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mspace width="4pt" /> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">}</mo> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Along the way, a Hardy–Littlewood-type theorem for general domains, which is of independent interests, is established.</p>